About the project

I am so excited to get an opportunity to be into R. I am expecting to learn how to hand data and visulaze the results of analysis through R. One of my colleagues had let me know about this course.

The URL for my git repository as follow:

https://github.com/Donggyu-Kam/IODS-project


Regression and Model validation (Excercise 2)

The data is the student’s attitude toward statistics.The data consists of information about students’ ages, students’ gender, three approaches, students’ attitude toward statistics and exam points. The oberservations whare the exam points are zeor were removed from data. And thus, the data has 166 observations and 7 variables. You can figure out three variables to affect the exam points through linear regression models and finally you will figure out the most significant variable to exampoints

#Before handling data, you should install two packages:“ggplot2” and“GGally”.

1.To read and explore data

  • Read data and look into the structure and dimension of data
  • Initialize plot with data and aesthetic mapping
    • p1 <- ggplot(df, aes(x = attitude,col=gender, y = points))
  • Define the visualization type (points)
    • p2 <- p1 + geom_point()
  • Draw the plot
    • p2
  • Add a regression line
    • p3 <- p2 + geom_smooth(method = “lm”)
  • Add a main title and draw the plot
    • p4 <- p3+ggtitle(“<Student’s attitude versus exam points>”) &
    • p4

2. To choose variables and create a regression model

By drawing plots illustrating the relationships between the variables and the normality of the erros distributions, it will find out what variables are closely related to exam points.

  • Draw a scatter plot matrix of the variables in learning2014
    • p <- ggpairs(df, mapping = aes(col=gender, alpha = 0.3), lower = list(combo = wrap(“facethist”, bins = 20)))
  • Plotting
    • p

According to the plot “p”, three variables “attitude”, “stra” and “surf” were chosen due to the strongest correlation value as explanatory variables.

  • Fit a linear model to each variable
    • qplot(attitude, points, data = df) + geom_smooth(method = “lm”)
    • qplot(stra, points, data=df)+ geom_smooth(method =“lm”)
    • qplot(surf, points, data=df)+ geom_smooth(method=“lm”)

After fitting each variable to a linear regression modle, “attitude” shows postive relationship but the others show week or negative relationships. So, the linear regression model is re-fitted with three variables together.

  • Fit a linear modewl with three variables together
    • mymodel<-lm(points~attitude+stra+surf, data=df)
  • Take a look at the summary of the model
    • summary(mymodel)

In this model, “Stra” and “Surf” are not not significantly related to the exam points. So, the model is re-fitted again witout the two variables

  • Fit the model again with the “attitude”

    • mymodel2<- lm(points ~ attitude, data=df)
    • summary(mymodel2)
    • par(mfrow = c(2,2))
    • plot(mymodel2, which=c(1,2,5)) - only take looks at the relation between residuals and fitted values, the normal QQplot, and the leverage of observations.

3. Result of regression model analysis

By analyzing the residuals and errors of the model, we can test the model validation. The variance of errors in residulas over fitted values looks constant without any patterns but there are 4 outliers indicating that the model only with “attitude” cannot explain the 4 ouliters of exam points. In the normality of errors in QQ plot, the errors are normaly distributed. In leverage of observations, standardized residuals are scattered around 0 and there seems no impact of outliers


Losistic Regression (Excercise 3)

The data is about student acheivement in secondary education of two Portugues schools. Two datasets are available for the performance in two subjects: Math and Portuguese language.

P. Cortez and A. Silva. Using Data Mining to Predict Secondary School Student Performance. In A. Brito and J. Teixeira Eds., Proceedings of 5th FUture BUsiness TEChnology Conference (FUBUTEC 2008) pp. 5-12, Porto, Portugal, April, 2008, EUROSIS, ISBN 978-9077381-39-7.[link]http://www3.dsi.uminho.pt/pcortez/student.pdf

1. Retreiving data

the data is about student acheivement in secondary education of two Portugues schools according to the student grades, demographic and school-related features. Two datasets are available for the performance in two subjects: Math and Portuguese language. the datasets consists of 35 variables and 382 observations.I firstly beging with installing some packages helps analysis. And thus, we can take a glimpse at the datasets and what variables are available through column names on the datasets.

library(dplyr);library(ggplot2);library(tidyr);library(boot)
## 
## Attaching package: 'dplyr'
## The following objects are masked from 'package:stats':
## 
##     filter, lag
## The following objects are masked from 'package:base':
## 
##     intersect, setdiff, setequal, union
alc<-read.csv("Z:\\IODS-project\\data\\alc.csv")
glimpse(alc)
## Observations: 382
## Variables: 36
## $ X          <int> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, ...
## $ school     <fct> GP, GP, GP, GP, GP, GP, GP, GP, GP, GP, GP, GP, GP,...
## $ sex        <fct> F, F, F, F, F, M, M, F, M, M, F, F, M, M, M, F, F, ...
## $ age        <int> 18, 17, 15, 15, 16, 16, 16, 17, 15, 15, 15, 15, 15,...
## $ address    <fct> U, U, U, U, U, U, U, U, U, U, U, U, U, U, U, U, U, ...
## $ famsize    <fct> GT3, GT3, LE3, GT3, GT3, LE3, LE3, GT3, LE3, GT3, G...
## $ Pstatus    <fct> A, T, T, T, T, T, T, A, A, T, T, T, T, T, A, T, T, ...
## $ Medu       <int> 4, 1, 1, 4, 3, 4, 2, 4, 3, 3, 4, 2, 4, 4, 2, 4, 4, ...
## $ Fedu       <int> 4, 1, 1, 2, 3, 3, 2, 4, 2, 4, 4, 1, 4, 3, 2, 4, 4, ...
## $ Mjob       <fct> at_home, at_home, at_home, health, other, services,...
## $ Fjob       <fct> teacher, other, other, services, other, other, othe...
## $ reason     <fct> course, course, other, home, home, reputation, home...
## $ nursery    <fct> yes, no, yes, yes, yes, yes, yes, yes, yes, yes, ye...
## $ internet   <fct> no, yes, yes, yes, no, yes, yes, no, yes, yes, yes,...
## $ guardian   <fct> mother, father, mother, mother, father, mother, mot...
## $ traveltime <int> 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 1, 1, ...
## $ studytime  <int> 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 3, 1, 2, 3, 1, 3, ...
## $ failures   <int> 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
## $ schoolsup  <fct> yes, no, yes, no, no, no, no, yes, no, no, no, no, ...
## $ famsup     <fct> no, yes, no, yes, yes, yes, no, yes, yes, yes, yes,...
## $ paid       <fct> no, no, yes, yes, yes, yes, no, no, yes, yes, yes, ...
## $ activities <fct> no, no, no, yes, no, yes, no, no, no, yes, no, yes,...
## $ higher     <fct> yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, y...
## $ romantic   <fct> no, no, no, yes, no, no, no, no, no, no, no, no, no...
## $ famrel     <int> 4, 5, 4, 3, 4, 5, 4, 4, 4, 5, 3, 5, 4, 5, 4, 4, 3, ...
## $ freetime   <int> 3, 3, 3, 2, 3, 4, 4, 1, 2, 5, 3, 2, 3, 4, 5, 4, 2, ...
## $ goout      <int> 4, 3, 2, 2, 2, 2, 4, 4, 2, 1, 3, 2, 3, 3, 2, 4, 3, ...
## $ Dalc       <int> 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
## $ Walc       <int> 1, 1, 3, 1, 2, 2, 1, 1, 1, 1, 2, 1, 3, 2, 1, 2, 2, ...
## $ health     <int> 3, 3, 3, 5, 5, 5, 3, 1, 1, 5, 2, 4, 5, 3, 3, 2, 2, ...
## $ absences   <int> 6, 4, 10, 2, 4, 10, 0, 6, 0, 0, 0, 4, 2, 2, 0, 4, 6...
## $ G1         <int> 5, 5, 7, 15, 6, 15, 12, 6, 16, 14, 10, 10, 14, 10, ...
## $ G2         <int> 6, 5, 8, 14, 10, 15, 12, 5, 18, 15, 8, 12, 14, 10, ...
## $ G3         <int> 6, 6, 10, 15, 10, 15, 11, 6, 19, 15, 9, 12, 14, 11,...
## $ alc_use    <dbl> 1.0, 1.0, 2.5, 1.0, 1.5, 1.5, 1.0, 1.0, 1.0, 1.0, 1...
## $ high_use   <lgl> TRUE, TRUE, FALSE, TRUE, TRUE, TRUE, TRUE, TRUE, TR...
gather(alc)%>% glimpse
## Warning: attributes are not identical across measure variables;
## they will be dropped
## Observations: 13,752
## Variables: 2
## $ key   <chr> "X", "X", "X", "X", "X", "X", "X", "X", "X", "X", "X", "...
## $ value <chr> "1", "2", "3", "4", "5", "6", "7", "8", "9", "10", "11",...
gather(alc)%>%ggplot(aes(value))+facet_wrap("key", scales="free")+geom_bar()
## Warning: attributes are not identical across measure variables;
## they will be dropped

2. Hypothesis

In order to study the relationship between high and low alcohol consumptions and variables, i am going to work folloing to hypothesis;

  1. Gender - Males consume more alcohol than females.
  2. G1 - First period grade decreases with alcohol consumption.
  3. G2 - Second period grade decreases with alcohol consumption.
  4. G3 - Third period grade decreases with alcohol consumption.

I will take a look at each relationship followring to the hypothesis one by one p<-ggpairs(alc, mapping = aes(col=sex, alpha = 0.3), lower = list(combo = wrap(“facethist”, bins = 20))) p

Hypothesis 1 - Gender

g1 <- ggplot(data = alc, aes(x = alc_use, fill = sex))
g1 + geom_bar()+facet_wrap("sex") + ggtitle("alohol consumption by sex")

g11<-ggplot(data=alc, aes(x=high_use, fill=sex))
g11+geom_bar()+facet_wrap("sex") + ggtitle("high and low alcohol consumption by sex")

Hypothesis 2 - G1

g2 <- ggplot(data=alc, aes(x=high_use, y= G1, col=sex))
g2+geom_boxplot()+ggtitle("G1 by alcohol consumption and sex")

Hypothesis 3 - G2

g3<-ggplot(alc, aes(x=high_use, y=G2, col=sex))
g3+geom_boxplot()+ggtitle("G2 by acohol consumption and sex")

Hypothesis 4 - Absences

g4<-ggplot(alc, aes(x=high_use, y=G3, col=sex))
g4+geom_boxplot()+ggtitle("G3 by alcohol consumption and sex")

Interim result

As a result of exploration of relationship with 4 variable to the high and low alcohol relationship, it seems that femailes consume more alcohol than males do. However, it is presumed that the more males consum alcohol, the less grades males achieve.

3.Lostistic regression

From now, in order to interpret relationship between variables presumably relating to alcohol consumptions and alcohol consumption, we will deal with logistaic regression.

m<-glm(high_use~sex+G1+G2+G3, data=alc, family="binomial")
summary(m)
## 
## Call:
## glm(formula = high_use ~ sex + G1 + G2 + G3, family = "binomial", 
##     data = alc)
## 
## Deviance Residuals: 
##     Min       1Q   Median       3Q      Max  
## -1.7050  -1.1393   0.8236   1.0275   1.4369  
## 
## Coefficients:
##              Estimate Std. Error z value Pr(>|z|)    
## (Intercept) -0.124551   0.377648  -0.330    0.742    
## sexM        -0.929752   0.215442  -4.316 1.59e-05 ***
## G1           0.100421   0.063556   1.580    0.114    
## G2           0.005342   0.076244   0.070    0.944    
## G3          -0.032884   0.053600  -0.614    0.540    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 524.94  on 381  degrees of freedom
## Residual deviance: 502.43  on 377  degrees of freedom
## AIC: 512.43
## 
## Number of Fisher Scoring iterations: 4
coef(m)
##  (Intercept)         sexM           G1           G2           G3 
## -0.124550963 -0.929752246  0.100420779  0.005341903 -0.032884237
OR<-coef(m)%>%exp
OR
## (Intercept)        sexM          G1          G2          G3 
##   0.8828933   0.3946515   1.1056360   1.0053562   0.9676506
CI<-exp(confint(m))
## Waiting for profiling to be done...
CI
##                 2.5 %    97.5 %
## (Intercept) 0.4202206 1.8520181
## sexM        0.2575779 0.5999088
## G1          0.9774251 1.2553494
## G2          0.8649561 1.1681070
## G3          0.8686826 1.0737634
cbind(OR, CI)
##                    OR     2.5 %    97.5 %
## (Intercept) 0.8828933 0.4202206 1.8520181
## sexM        0.3946515 0.2575779 0.5999088
## G1          1.1056360 0.9774251 1.2553494
## G2          1.0053562 0.8649561 1.1681070
## G3          0.9676506 0.8686826 1.0737634

The model indicates taht males are more significantly related to high alcohol consumptions.However, the other 3 variables are not significantly related to high alcohol consumptions. In odds rates, the odds rates for G2 and G3 are more than 1.

Prediction power

probabilities<-predict(m, type="response")
alc<-mutate(alc,probability=probabilities)
alc<-mutate(alc,prediction=probability>0.5)
select(alc,G2,G3, high_use, probability, prediction)%>% tail(10)
##     G2 G3 high_use probability prediction
## 373  5  0    FALSE   0.3953044      FALSE
## 374  5  5    FALSE   0.3567518      FALSE
## 375  9 10    FALSE   0.6454027       TRUE
## 376  5  6    FALSE   0.5762451       TRUE
## 377  5  0     TRUE   0.6468232       TRUE
## 378  9  8     TRUE   0.5898627       TRUE
## 379  5  0     TRUE   0.6235593       TRUE
## 380  5  0     TRUE   0.6235593       TRUE
## 381 16 16    FALSE   0.4777423      FALSE
## 382 12 10    FALSE   0.4466009      FALSE

2x2 cross-tabulation and graphic representation

g<-ggplot(alc,(aes(x=probability, y=high_use, col=prediction)))
g+geom_point()

table(high_use=alc$high_use, prediction=alc$prediction)%>%prop.table()%>%addmargins()
##         prediction
## high_use     FALSE      TRUE       Sum
##    FALSE 0.2225131 0.2225131 0.4450262
##    TRUE  0.1492147 0.4057592 0.5549738
##    Sum   0.3717277 0.6282723 1.0000000

Exploring the training error

loss_func<-function(class, prob){
  n_wrong <-abs(class - prob) > 0.5
  mean(n_wrong)
}
loss_func(class = alc$high_use, prob = alc$probability)
## [1] 0.3717277

Through exploring the training error, the incorrection of prediction is almost 50%.

10-fold cross-validation (Bonus task)

cv<-cv.glm(data=alc, cost=loss_func, glmfit = m, K=10)
cv$delta[1]
## [1] 0.3795812

Eventually, the prediction error with the 4 variables is 40%.


Clustering and classification (Exercise 4)

library(MASS);library(tidyverse);library(corrplot)
## 
## Attaching package: 'MASS'
## The following object is masked from 'package:dplyr':
## 
##     select
## -- Attaching packages -------------------------------------------------------------------------------------- tidyverse 1.2.1 --
## v tibble  2.1.3     v purrr   0.3.3
## v readr   1.3.1     v stringr 1.4.0
## v tibble  2.1.3     v forcats 0.4.0
## -- Conflicts ----------------------------------------------------------------------------------------- tidyverse_conflicts() --
## x dplyr::filter() masks stats::filter()
## x dplyr::lag()    masks stats::lag()
## x MASS::select()  masks dplyr::select()
## corrplot 0.84 loaded
data("Boston")

The data has 506 observations and 14 variables. Data is about housing values in suburbs of Boston.

Explort dataset

str(Boston)
## 'data.frame':    506 obs. of  14 variables:
##  $ crim   : num  0.00632 0.02731 0.02729 0.03237 0.06905 ...
##  $ zn     : num  18 0 0 0 0 0 12.5 12.5 12.5 12.5 ...
##  $ indus  : num  2.31 7.07 7.07 2.18 2.18 2.18 7.87 7.87 7.87 7.87 ...
##  $ chas   : int  0 0 0 0 0 0 0 0 0 0 ...
##  $ nox    : num  0.538 0.469 0.469 0.458 0.458 0.458 0.524 0.524 0.524 0.524 ...
##  $ rm     : num  6.58 6.42 7.18 7 7.15 ...
##  $ age    : num  65.2 78.9 61.1 45.8 54.2 58.7 66.6 96.1 100 85.9 ...
##  $ dis    : num  4.09 4.97 4.97 6.06 6.06 ...
##  $ rad    : int  1 2 2 3 3 3 5 5 5 5 ...
##  $ tax    : num  296 242 242 222 222 222 311 311 311 311 ...
##  $ ptratio: num  15.3 17.8 17.8 18.7 18.7 18.7 15.2 15.2 15.2 15.2 ...
##  $ black  : num  397 397 393 395 397 ...
##  $ lstat  : num  4.98 9.14 4.03 2.94 5.33 ...
##  $ medv   : num  24 21.6 34.7 33.4 36.2 28.7 22.9 27.1 16.5 18.9 ...
summary(Boston)
##       crim                zn             indus            chas        
##  Min.   : 0.00632   Min.   :  0.00   Min.   : 0.46   Min.   :0.00000  
##  1st Qu.: 0.08204   1st Qu.:  0.00   1st Qu.: 5.19   1st Qu.:0.00000  
##  Median : 0.25651   Median :  0.00   Median : 9.69   Median :0.00000  
##  Mean   : 3.61352   Mean   : 11.36   Mean   :11.14   Mean   :0.06917  
##  3rd Qu.: 3.67708   3rd Qu.: 12.50   3rd Qu.:18.10   3rd Qu.:0.00000  
##  Max.   :88.97620   Max.   :100.00   Max.   :27.74   Max.   :1.00000  
##       nox               rm             age              dis        
##  Min.   :0.3850   Min.   :3.561   Min.   :  2.90   Min.   : 1.130  
##  1st Qu.:0.4490   1st Qu.:5.886   1st Qu.: 45.02   1st Qu.: 2.100  
##  Median :0.5380   Median :6.208   Median : 77.50   Median : 3.207  
##  Mean   :0.5547   Mean   :6.285   Mean   : 68.57   Mean   : 3.795  
##  3rd Qu.:0.6240   3rd Qu.:6.623   3rd Qu.: 94.08   3rd Qu.: 5.188  
##  Max.   :0.8710   Max.   :8.780   Max.   :100.00   Max.   :12.127  
##       rad              tax           ptratio          black       
##  Min.   : 1.000   Min.   :187.0   Min.   :12.60   Min.   :  0.32  
##  1st Qu.: 4.000   1st Qu.:279.0   1st Qu.:17.40   1st Qu.:375.38  
##  Median : 5.000   Median :330.0   Median :19.05   Median :391.44  
##  Mean   : 9.549   Mean   :408.2   Mean   :18.46   Mean   :356.67  
##  3rd Qu.:24.000   3rd Qu.:666.0   3rd Qu.:20.20   3rd Qu.:396.23  
##  Max.   :24.000   Max.   :711.0   Max.   :22.00   Max.   :396.90  
##      lstat            medv      
##  Min.   : 1.73   Min.   : 5.00  
##  1st Qu.: 6.95   1st Qu.:17.02  
##  Median :11.36   Median :21.20  
##  Mean   :12.65   Mean   :22.53  
##  3rd Qu.:16.95   3rd Qu.:25.00  
##  Max.   :37.97   Max.   :50.00
pairs(Boston)

Correlate all variables

cor_matrix<-cor(Boston)
cor_matrix%>%round(2)
##          crim    zn indus  chas   nox    rm   age   dis   rad   tax
## crim     1.00 -0.20  0.41 -0.06  0.42 -0.22  0.35 -0.38  0.63  0.58
## zn      -0.20  1.00 -0.53 -0.04 -0.52  0.31 -0.57  0.66 -0.31 -0.31
## indus    0.41 -0.53  1.00  0.06  0.76 -0.39  0.64 -0.71  0.60  0.72
## chas    -0.06 -0.04  0.06  1.00  0.09  0.09  0.09 -0.10 -0.01 -0.04
## nox      0.42 -0.52  0.76  0.09  1.00 -0.30  0.73 -0.77  0.61  0.67
## rm      -0.22  0.31 -0.39  0.09 -0.30  1.00 -0.24  0.21 -0.21 -0.29
## age      0.35 -0.57  0.64  0.09  0.73 -0.24  1.00 -0.75  0.46  0.51
## dis     -0.38  0.66 -0.71 -0.10 -0.77  0.21 -0.75  1.00 -0.49 -0.53
## rad      0.63 -0.31  0.60 -0.01  0.61 -0.21  0.46 -0.49  1.00  0.91
## tax      0.58 -0.31  0.72 -0.04  0.67 -0.29  0.51 -0.53  0.91  1.00
## ptratio  0.29 -0.39  0.38 -0.12  0.19 -0.36  0.26 -0.23  0.46  0.46
## black   -0.39  0.18 -0.36  0.05 -0.38  0.13 -0.27  0.29 -0.44 -0.44
## lstat    0.46 -0.41  0.60 -0.05  0.59 -0.61  0.60 -0.50  0.49  0.54
## medv    -0.39  0.36 -0.48  0.18 -0.43  0.70 -0.38  0.25 -0.38 -0.47
##         ptratio black lstat  medv
## crim       0.29 -0.39  0.46 -0.39
## zn        -0.39  0.18 -0.41  0.36
## indus      0.38 -0.36  0.60 -0.48
## chas      -0.12  0.05 -0.05  0.18
## nox        0.19 -0.38  0.59 -0.43
## rm        -0.36  0.13 -0.61  0.70
## age        0.26 -0.27  0.60 -0.38
## dis       -0.23  0.29 -0.50  0.25
## rad        0.46 -0.44  0.49 -0.38
## tax        0.46 -0.44  0.54 -0.47
## ptratio    1.00 -0.18  0.37 -0.51
## black     -0.18  1.00 -0.37  0.33
## lstat      0.37 -0.37  1.00 -0.74
## medv      -0.51  0.33 -0.74  1.00
corrplot(cor_matrix, method="circle", type="upper", cl.pos="b", tl.pos="d", tl.cex=0.6)

There is strongly positive correlation bewtween “rad” and “tax” and stongly negative correlation between “age” and “dis” and between “lstat” and “medv”.

Standardized data

boston_scaled<-scale(Boston)
summary(boston_scaled)
##       crim                 zn               indus        
##  Min.   :-0.419367   Min.   :-0.48724   Min.   :-1.5563  
##  1st Qu.:-0.410563   1st Qu.:-0.48724   1st Qu.:-0.8668  
##  Median :-0.390280   Median :-0.48724   Median :-0.2109  
##  Mean   : 0.000000   Mean   : 0.00000   Mean   : 0.0000  
##  3rd Qu.: 0.007389   3rd Qu.: 0.04872   3rd Qu.: 1.0150  
##  Max.   : 9.924110   Max.   : 3.80047   Max.   : 2.4202  
##       chas              nox                rm               age         
##  Min.   :-0.2723   Min.   :-1.4644   Min.   :-3.8764   Min.   :-2.3331  
##  1st Qu.:-0.2723   1st Qu.:-0.9121   1st Qu.:-0.5681   1st Qu.:-0.8366  
##  Median :-0.2723   Median :-0.1441   Median :-0.1084   Median : 0.3171  
##  Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000  
##  3rd Qu.:-0.2723   3rd Qu.: 0.5981   3rd Qu.: 0.4823   3rd Qu.: 0.9059  
##  Max.   : 3.6648   Max.   : 2.7296   Max.   : 3.5515   Max.   : 1.1164  
##       dis               rad               tax             ptratio       
##  Min.   :-1.2658   Min.   :-0.9819   Min.   :-1.3127   Min.   :-2.7047  
##  1st Qu.:-0.8049   1st Qu.:-0.6373   1st Qu.:-0.7668   1st Qu.:-0.4876  
##  Median :-0.2790   Median :-0.5225   Median :-0.4642   Median : 0.2746  
##  Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000  
##  3rd Qu.: 0.6617   3rd Qu.: 1.6596   3rd Qu.: 1.5294   3rd Qu.: 0.8058  
##  Max.   : 3.9566   Max.   : 1.6596   Max.   : 1.7964   Max.   : 1.6372  
##      black             lstat              medv        
##  Min.   :-3.9033   Min.   :-1.5296   Min.   :-1.9063  
##  1st Qu.: 0.2049   1st Qu.:-0.7986   1st Qu.:-0.5989  
##  Median : 0.3808   Median :-0.1811   Median :-0.1449  
##  Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000  
##  3rd Qu.: 0.4332   3rd Qu.: 0.6024   3rd Qu.: 0.2683  
##  Max.   : 0.4406   Max.   : 3.5453   Max.   : 2.9865

Covert matrix data to dafaframe

class(boston_scaled)
## [1] "matrix"
boston_scaled<-as.data.frame(boston_scaled)

Create a catergorical variable with the variable “Crime”

summary("crim")
##    Length     Class      Mode 
##         1 character character
bins<-quantile(boston_scaled$crim)
bins
##           0%          25%          50%          75%         100% 
## -0.419366929 -0.410563278 -0.390280295  0.007389247  9.924109610
crime<-cut(boston_scaled$crim, breaks=bins, include.lowest=TRUE, label=c("low","med_low","med_high", "high"))
crime
##   [1] low      low      low      low      low      low      med_low 
##   [8] med_low  med_low  med_low  med_low  med_low  med_low  med_high
##  [15] med_high med_high med_high med_high med_high med_high med_high
##  [22] med_high med_high med_high med_high med_high med_high med_high
##  [29] med_high med_high med_high med_high med_high med_high med_high
##  [36] low      med_low  low      med_low  low      low      med_low 
##  [43] med_low  med_low  med_low  med_low  med_low  med_low  med_low 
##  [50] med_low  med_low  low      low      low      low      low     
##  [57] low      low      med_low  med_low  med_low  med_low  med_low 
##  [64] med_low  low      low      low      low      med_low  med_low 
##  [71] med_low  med_low  med_low  med_low  low      med_low  med_low 
##  [78] med_low  low      med_low  low      low      low      low     
##  [85] low      low      low      low      low      low      low     
##  [92] low      low      low      low      med_low  med_low  med_low 
##  [99] low      low      med_low  med_low  med_low  med_low  med_low 
## [106] med_low  med_low  med_low  med_low  med_high med_low  med_low 
## [113] med_low  med_low  med_low  med_low  med_low  med_low  med_low 
## [120] med_low  low      low      med_low  med_low  med_low  med_low 
## [127] med_high med_high med_high med_high med_high med_high med_high
## [134] med_high med_high med_high med_high med_high med_low  med_high
## [141] med_high med_high med_high high     med_high med_high med_high
## [148] med_high med_high med_high med_high med_high med_high med_high
## [155] med_high med_high med_high med_high med_high med_high med_high
## [162] med_high med_high med_high med_high med_high med_high med_high
## [169] med_high med_high med_high med_high med_low  med_low  med_low 
## [176] low      low      low      low      low      low      low     
## [183] med_low  med_low  med_low  low      low      low      med_low 
## [190] med_low  med_low  low      med_low  low      low      low     
## [197] low      low      low      low      low      low      low     
## [204] low      low      med_low  med_low  med_low  med_low  med_high
## [211] med_low  med_high med_low  med_low  med_high med_low  low     
## [218] low      med_low  med_low  med_high med_high med_high med_high
## [225] med_high med_high med_high med_high med_high med_high med_high
## [232] med_high med_high med_high med_high med_high med_high med_high
## [239] med_low  med_low  med_low  med_low  med_low  med_low  med_low 
## [246] med_low  med_high med_low  med_low  med_low  med_low  med_low 
## [253] med_low  med_high low      low      low      med_high med_high
## [260] med_high med_high med_high med_high med_high med_high med_high
## [267] med_high med_high med_high med_low  med_high med_low  med_low 
## [274] med_low  low      med_low  med_low  low      low      med_low 
## [281] low      low      low      low      low      low      low     
## [288] low      low      low      low      low      low      med_low 
## [295] low      med_low  low      med_low  low      low      low     
## [302] low      med_low  med_low  low      low      low      low     
## [309] med_high med_high med_high med_high med_high med_high med_high
## [316] med_low  med_high med_low  med_high med_high med_low  med_low 
## [323] med_high med_high med_high med_low  med_high med_low  low     
## [330] low      low      low      low      low      low      low     
## [337] low      low      low      low      low      low      low     
## [344] low      low      low      low      low      low      low     
## [351] low      low      low      low      low      med_low  high    
## [358] high     high     high     high     high     high     high    
## [365] med_high high     high     high     high     high     high    
## [372] high     high     high     high     high     high     high    
## [379] high     high     high     high     high     high     high    
## [386] high     high     high     high     high     high     high    
## [393] high     high     high     high     high     high     high    
## [400] high     high     high     high     high     high     high    
## [407] high     high     high     high     high     high     high    
## [414] high     high     high     high     high     high     high    
## [421] high     high     high     high     high     high     high    
## [428] high     high     high     high     high     high     high    
## [435] high     high     high     high     high     high     high    
## [442] high     high     high     high     high     high     high    
## [449] high     high     high     high     high     high     high    
## [456] high     high     high     high     high     high     high    
## [463] high     high     high     med_high high     high     high    
## [470] high     high     high     med_high high     high     high    
## [477] high     high     high     high     high     high     high    
## [484] med_high med_high med_high high     high     med_low  med_low 
## [491] med_low  med_low  med_low  med_low  med_high med_low  med_high
## [498] med_high med_low  med_low  med_low  low      low      low     
## [505] med_low  low     
## Levels: low med_low med_high high
table(crime)
## crime
##      low  med_low med_high     high 
##      127      126      126      127

Remove original “crim” from the dataset and add the new categorial variable

boston_scaled<-dplyr::select(boston_scaled,-crim)
boston_scaled
##              zn       indus       chas         nox           rm
## 1    0.28454827 -1.28663623 -0.2723291 -0.14407485  0.413262920
## 2   -0.48724019 -0.59279438 -0.2723291 -0.73953036  0.194082387
## 3   -0.48724019 -0.59279438 -0.2723291 -0.73953036  1.281445551
## 4   -0.48724019 -1.30558569 -0.2723291 -0.83445805  1.015297761
## 5   -0.48724019 -1.30558569 -0.2723291 -0.83445805  1.227362043
## 6   -0.48724019 -1.30558569 -0.2723291 -0.83445805  0.206891639
## 7    0.04872402 -0.47618230 -0.2723291 -0.26489191 -0.388026950
## 8    0.04872402 -0.47618230 -0.2723291 -0.26489191 -0.160306916
## 9    0.04872402 -0.47618230 -0.2723291 -0.26489191 -0.930285282
## 10   0.04872402 -0.47618230 -0.2723291 -0.26489191 -0.399412952
## 11   0.04872402 -0.47618230 -0.2723291 -0.26489191  0.131459378
## 12   0.04872402 -0.47618230 -0.2723291 -0.26489191 -0.392296701
## 13   0.04872402 -0.47618230 -0.2723291 -0.26489191 -0.563086727
## 14  -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.477691714
## 15  -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.268473932
## 16  -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.641365488
## 17  -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.497617217
## 18  -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.419338455
## 19  -0.48724019 -0.43682573 -0.2723291 -0.14407485 -1.179354069
## 20  -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.793653261
## 21  -0.48724019 -0.43682573 -0.2723291 -0.14407485 -1.017103545
## 22  -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.454919710
## 23  -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.203004422
## 24  -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.671253743
## 25  -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.513272969
## 26  -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.975829289
## 27  -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.671253743
## 28  -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.338213193
## 29  -0.48724019 -0.43682573 -0.2723291 -0.14407485  0.299402903
## 30  -0.48724019 -0.43682573 -0.2723291 -0.14407485  0.554164692
## 31  -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.813578764
## 32  -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.302631937
## 33  -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.476268464
## 34  -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.830657767
## 35  -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.268473932
## 36  -0.48724019 -0.75459363 -0.2723291 -0.48063666 -0.500463717
## 37  -0.48724019 -0.75459363 -0.2723291 -0.48063666 -0.631402737
## 38  -0.48724019 -0.75459363 -0.2723291 -0.48063666 -0.618593485
## 39  -0.48724019 -0.75459363 -0.2723291 -0.48063666 -0.453496460
## 40   2.72854505 -1.19334657 -0.2723291 -1.09335175  0.441727925
## 41   2.72854505 -1.19334657 -0.2723291 -1.09335175  1.052302267
## 42  -0.48724019 -0.61611679 -0.2723291 -0.92075595  0.690796712
## 43  -0.48724019 -0.61611679 -0.2723291 -0.92075595 -0.164576667
## 44  -0.48724019 -0.61611679 -0.2723291 -0.92075595 -0.104800158
## 45  -0.48724019 -0.61611679 -0.2723291 -0.92075595 -0.306901688
## 46  -0.48724019 -0.61611679 -0.2723291 -0.92075595 -0.857699521
## 47  -0.48724019 -0.61611679 -0.2723291 -0.92075595 -0.709681499
## 48  -0.48724019 -0.61611679 -0.2723291 -0.92075595 -0.362408446
## 49  -0.48724019 -0.61611679 -0.2723291 -0.92075595 -1.260479332
## 50  -0.48724019 -0.61611679 -0.2723291 -0.92075595 -0.971559538
## 51   0.41317968 -0.80123846 -0.2723291 -0.99842406 -0.457766211
## 52   0.41317968 -0.80123846 -0.2723291 -0.99842406 -0.241432178
## 53   0.41317968 -0.80123846 -0.2723291 -0.99842406  0.322174907
## 54   0.41317968 -0.80123846 -0.2723291 -0.99842406 -0.407952453
## 55   2.72854505 -1.04029322 -0.2723291 -1.24868797 -0.564509977
## 56   3.37170210 -1.44552019 -0.2723291 -1.30909650  1.372533565
## 57   3.15731641 -1.51548743 -0.2723291 -1.24868797  0.139998879
## 58   3.80047346 -1.43094368 -0.2723291 -1.24005818  0.756266222
## 59   0.58468822 -0.87557866 -0.2723291 -0.87760700 -0.198734672
## 60   0.58468822 -0.87557866 -0.2723291 -0.87760700 -0.509003218
## 61   0.58468822 -0.87557866 -0.2723291 -0.87760700 -0.773727758
## 62   0.58468822 -0.87557866 -0.2723291 -0.87760700 -0.453496460
## 63   0.58468822 -0.87557866 -0.2723291 -0.87760700  0.243896145
## 64   0.58468822 -0.87557866 -0.2723291 -0.87760700  0.679410711
## 65   0.26310970 -1.42219777 -0.2723291 -1.19604625  1.166162284
## 66   2.94293073 -1.13212523 -0.2723291 -1.35224545  0.007636609
## 67   2.94293073 -1.13212523 -0.2723291 -1.35224545 -0.708258248
## 68   0.04872402 -0.73855947 -0.2723291 -1.25731776 -0.578742479
## 69   0.04872402 -0.73855947 -0.2723291 -1.25731776 -0.982945540
## 70   0.04872402 -0.73855947 -0.2723291 -1.25731776 -0.568779727
## 71  -0.48724019 -0.04763292 -0.2723291 -1.22279860  0.188389387
## 72  -0.48724019 -0.04763292 -0.2723291 -1.22279860 -0.460612711
## 73  -0.48724019 -0.04763292 -0.2723291 -1.22279860 -0.312594689
## 74  -0.48724019 -0.04763292 -0.2723291 -1.22279860 -0.056409650
## 75  -0.48724019  0.24681257 -0.2723291 -1.01568364 -0.016558644
## 76  -0.48724019  0.24681257 -0.2723291 -1.01568364  0.001943608
## 77  -0.48724019  0.24681257 -0.2723291 -1.01568364 -0.008019143
## 78  -0.48724019  0.24681257 -0.2723291 -1.01568364 -0.205850923
## 79  -0.48724019  0.24681257 -0.2723291 -1.01568364 -0.074911903
## 80  -0.48724019  0.24681257 -0.2723291 -1.01568364 -0.584435480
## 81   0.58468822 -0.91493524 -0.2723291 -1.11061133  0.629596953
## 82   0.58468822 -0.91493524 -0.2723291 -1.11061133  0.475885930
## 83   0.58468822 -0.91493524 -0.2723291 -1.11061133  0.024715612
## 84   0.58468822 -0.91493524 -0.2723291 -1.11061133 -0.167423167
## 85  -0.48724019 -0.96886832 -0.2723291 -0.91212616  0.148538381
## 86  -0.48724019 -0.96886832 -0.2723291 -0.91212616  0.491541682
## 87  -0.48724019 -0.96886832 -0.2723291 -0.91212616 -0.383757200
## 88  -0.48724019 -0.96886832 -0.2723291 -0.91212616 -0.232892677
## 89  -0.48724019 -1.12629463 -0.2723291 -0.56693456  1.028107013
## 90  -0.48724019 -1.12629463 -0.2723291 -0.56693456  1.130581029
## 91  -0.48724019 -1.12629463 -0.2723291 -0.56693456  0.188389387
## 92  -0.48724019 -1.12629463 -0.2723291 -0.56693456  0.171310384
## 93   0.71331963  0.56895343 -0.2723291 -0.78267931  0.223970642
## 94   0.71331963  0.56895343 -0.2723291 -0.78267931 -0.104800158
## 95   0.71331963  0.56895343 -0.2723291 -0.78267931 -0.050716649
## 96  -0.48724019 -1.20209248 -0.2723291 -0.94664532  0.484425431
## 97  -0.48724019 -1.20209248 -0.2723291 -0.94664532 -0.173116168
## 98  -0.48724019 -1.20209248 -0.2723291 -0.94664532  2.539598741
## 99  -0.48724019 -1.20209248 -0.2723291 -0.94664532  2.185209437
## 100 -0.48724019 -1.20209248 -0.2723291 -0.94664532  1.610216351
## 101 -0.48724019 -0.37560439 -0.2723291 -0.29941107  0.629596953
## 102 -0.48724019 -0.37560439 -0.2723291 -0.29941107  0.706452465
## 103 -0.48724019 -0.37560439 -0.2723291 -0.29941107  0.171310384
## 104 -0.48724019 -0.37560439 -0.2723291 -0.29941107 -0.210120673
## 105 -0.48724019 -0.37560439 -0.2723291 -0.29941107 -0.167423167
## 106 -0.48724019 -0.37560439 -0.2723291 -0.29941107 -0.617170235
## 107 -0.48724019 -0.37560439 -0.2723291 -0.29941107 -0.638518988
## 108 -0.48724019 -0.37560439 -0.2723291 -0.29941107 -0.224353176
## 109 -0.48724019 -0.37560439 -0.2723291 -0.29941107  0.269514649
## 110 -0.48724019 -0.37560439 -0.2723291 -0.29941107 -0.079181654
## 111 -0.48724019 -0.37560439 -0.2723291 -0.29941107 -0.127572161
## 112 -0.48724019 -0.16424500 -0.2723291 -0.06640675  0.612517950
## 113 -0.48724019 -0.16424500 -0.2723291 -0.06640675 -0.528928721
## 114 -0.48724019 -0.16424500 -0.2723291 -0.06640675 -0.274166933
## 115 -0.48724019 -0.16424500 -0.2723291 -0.06640675 -0.043600398
## 116 -0.48724019 -0.16424500 -0.2723291 -0.06640675 -0.507579968
## 117 -0.48724019 -0.16424500 -0.2723291 -0.06640675 -0.154613915
## 118 -0.48724019 -0.16424500 -0.2723291 -0.06640675 -0.375217698
## 119 -0.48724019 -0.16424500 -0.2723291 -0.06640675 -0.587281980
## 120 -0.48724019 -0.16424500 -0.2723291 -0.06640675 -0.787960260
## 121 -0.48724019  2.11552109 -0.2723291  0.22700611 -0.590128481
## 122 -0.48724019  2.11552109 -0.2723291  0.22700611 -0.399412952
## 123 -0.48724019  2.11552109 -0.2723291  0.22700611 -0.460612711
## 124 -0.48724019  2.11552109 -0.2723291  0.22700611 -0.610053984
## 125 -0.48724019  2.11552109 -0.2723291  0.22700611 -0.577319229
## 126 -0.48724019  2.11552109 -0.2723291  0.22700611 -0.425031456
## 127 -0.48724019  2.11552109 -0.2723291  0.22700611 -0.955903786
## 128 -0.48724019  1.56744433 -0.2723291  0.59808708 -0.842043769
## 129 -0.48724019  1.56744433 -0.2723291  0.59808708  0.208314890
## 130 -0.48724019  1.56744433 -0.2723291  0.59808708 -0.921745781
## 131 -0.48724019  1.56744433 -0.2723291  0.59808708  0.246742645
## 132 -0.48724019  1.56744433 -0.2723291  0.59808708  0.058873617
## 133 -0.48724019  1.56744433 -0.2723291  0.59808708  0.124343127
## 134 -0.48724019  1.56744433 -0.2723291  0.59808708 -0.658444491
## 135 -0.48724019  1.56744433 -0.2723291  0.59808708 -0.750955755
## 136 -0.48724019  1.56744433 -0.2723291  0.59808708  0.071682869
## 137 -0.48724019  1.56744433 -0.2723291  0.59808708 -0.487654465
## 138 -0.48724019  1.56744433 -0.2723291  0.59808708  0.241049645
## 139 -0.48724019  1.56744433 -0.2723291  0.59808708 -0.608630733
## 140 -0.48724019  1.56744433 -0.2723291  0.59808708 -0.190195170
## 141 -0.48724019  1.56744433 -0.2723291  0.59808708 -0.157460416
## 142 -0.48724019  1.56744433 -0.2723291  0.59808708 -1.801314413
## 143 -0.48724019  1.23072696  3.6647712  2.72964520 -1.254786331
## 144 -0.48724019  1.23072696 -0.2723291  2.72964520 -1.162275067
## 145 -0.48724019  1.23072696 -0.2723291  2.72964520 -1.966411438
## 146 -0.48724019  1.23072696 -0.2723291  2.72964520 -0.220083425
## 147 -0.48724019  1.23072696 -0.2723291  2.72964520 -0.934555033
## 148 -0.48724019  1.23072696 -0.2723291  2.72964520 -1.933676683
## 149 -0.48724019  1.23072696 -0.2723291  2.72964520 -1.563631627
## 150 -0.48724019  1.23072696 -0.2723291  2.72964520 -0.978675789
## 151 -0.48724019  1.23072696 -0.2723291  2.72964520 -0.231469427
## 152 -0.48724019  1.23072696 -0.2723291  2.72964520 -1.253363081
## 153 -0.48724019  1.23072696  3.6647712  2.72964520 -1.811277165
## 154 -0.48724019  1.23072696 -0.2723291  2.72964520 -0.819271765
## 155 -0.48724019  1.23072696  3.6647712  2.72964520 -0.221506675
## 156 -0.48724019  1.23072696  3.6647712  2.72964520 -0.188771920
## 157 -0.48724019  1.23072696 -0.2723291  2.72964520 -1.441232109
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## 381 -0.48724019  1.01499462 -0.2723291  1.00368721  0.972600255
## 382 -0.48724019  1.01499462 -0.2723291  1.00368721  0.370565414
## 383 -0.48724019  1.01499462 -0.2723291  1.25395112 -1.065494052
## 384 -0.48724019  1.01499462 -0.2723291  1.25395112 -1.088266056
## 385 -0.48724019  1.01499462 -0.2723291  1.25395112 -2.727850303
## 386 -0.48724019  1.01499462 -0.2723291  1.25395112 -1.434115858
## 387 -0.48724019  1.01499462 -0.2723291  1.25395112 -2.323647242
## 388 -0.48724019  1.01499462 -0.2723291  1.25395112 -1.828356167
## 389 -0.48724019  1.01499462 -0.2723291  1.25395112 -1.999146193
## 390 -0.48724019  1.01499462 -0.2723291  1.25395112 -1.273288584
## 391 -0.48724019  1.01499462 -0.2723291  1.25395112 -0.813578764
## 392 -0.48724019  1.01499462 -0.2723291  1.25395112 -0.332520192
## 393 -0.48724019  1.01499462 -0.2723291  1.25395112 -1.777119159
## 394 -0.48724019  1.01499462 -0.2723291  1.19354259 -0.130418661
## 395 -0.48724019  1.01499462 -0.2723291  1.19354259 -0.565933227
## 396 -0.48724019  1.01499462 -0.2723291  1.19354259  0.265244898
## 397 -0.48724019  1.01499462 -0.2723291  1.19354259  0.171310384
## 398 -0.48724019  1.01499462 -0.2723291  1.19354259 -0.765188257
## 399 -0.48724019  1.01499462 -0.2723291  1.19354259 -1.183623820
## 400 -0.48724019  1.01499462 -0.2723291  1.19354259 -0.615746985
## 401 -0.48724019  1.01499462 -0.2723291  1.19354259 -0.423608206
## 402 -0.48724019  1.01499462 -0.2723291  1.19354259  0.083068871
## 403 -0.48724019  1.01499462 -0.2723291  1.19354259  0.169887134
## 404 -0.48724019  1.01499462 -0.2723291  1.19354259 -1.331641842
## 405 -0.48724019  1.01499462 -0.2723291  1.19354259 -1.072610303
## 406 -0.48724019  1.01499462 -0.2723291  1.19354259 -0.856276271
## 407 -0.48724019  1.01499462 -0.2723291  0.90012973 -3.055197852
## 408 -0.48724019  1.01499462 -0.2723291  0.90012973 -0.963020037
## 409 -0.48724019  1.01499462 -0.2723291  0.36508275 -0.950210785
## 410 -0.48724019  1.01499462 -0.2723291  0.36508275  0.807503230
## 411 -0.48724019  1.01499462 -0.2723291  0.36508275 -0.750955755
## 412 -0.48724019  1.01499462 -0.2723291  0.36508275  0.529969438
## 413 -0.48724019  1.01499462 -0.2723291  0.36508275 -2.357805247
## 414 -0.48724019  1.01499462 -0.2723291  0.36508275 -1.607752384
## 415 -0.48724019  1.01499462 -0.2723291  1.19354259 -2.512939520
## 416 -0.48724019  1.01499462 -0.2723291  1.07272553  0.212584640
## 417 -0.48724019  1.01499462 -0.2723291  1.07272553  0.707875715
## 418 -0.48724019  1.01499462 -0.2723291  1.07272553 -1.395688102
## 419 -0.48724019  1.01499462 -0.2723291  1.07272553 -0.466305712
## 420 -0.48724019  1.01499462 -0.2723291  1.40928733  0.767652224
## 421 -0.48724019  1.01499462 -0.2723291  1.40928733  0.179849885
## 422 -0.48724019  1.01499462 -0.2723291  1.40928733 -0.396566452
## 423 -0.48724019  1.01499462 -0.2723291  0.51178918 -0.906090028
## 424 -0.48724019  1.01499462 -0.2723291  0.51178918 -0.258511181
## 425 -0.48724019  1.01499462 -0.2723291  0.25289548 -1.024219796
## 426 -0.48724019  1.01499462 -0.2723291  1.07272553 -0.553123975
## 427 -0.48724019  1.01499462 -0.2723291  0.25289548 -0.637095738
## 428 -0.48724019  1.01499462 -0.2723291  1.07272553 -0.117609410
## 429 -0.48724019  1.01499462 -0.2723291  1.07272553 -0.130418661
## 430 -0.48724019  1.01499462 -0.2723291  1.07272553  0.135729129
## 431 -0.48724019  1.01499462 -0.2723291  0.25289548  0.090185122
## 432 -0.48724019  1.01499462 -0.2723291  0.25289548  0.780461476
## 433 -0.48724019  1.01499462 -0.2723291  0.25289548  0.199775388
## 434 -0.48724019  1.01499462 -0.2723291  1.36613839  0.215431141
## 435 -0.48724019  1.01499462 -0.2723291  1.36613839 -0.109069908
## 436 -0.48724019  1.01499462 -0.2723291  1.59914271  0.490118432
## 437 -0.48724019  1.01499462 -0.2723291  1.59914271  0.251012396
## 438 -0.48724019  1.01499462 -0.2723291  1.59914271 -0.188771920
## 439 -0.48724019  1.01499462 -0.2723291  1.59914271 -0.497617217
## 440 -0.48724019  1.01499462 -0.2723291  1.59914271 -0.935978283
## 441 -0.48724019  1.01499462 -0.2723291  1.59914271 -0.664137492
## 442 -0.48724019  1.01499462 -0.2723291  1.59914271  0.172733634
## 443 -0.48724019  1.01499462 -0.2723291  1.59914271 -0.093414156
## 444 -0.48724019  1.01499462 -0.2723291  1.59914271  0.285170401
## 445 -0.48724019  1.01499462 -0.2723291  1.59914271 -0.612900484
## 446 -0.48724019  1.01499462 -0.2723291  1.59914271  0.248165896
## 447 -0.48724019  1.01499462 -0.2723291  1.59914271  0.080222370
## 448 -0.48724019  1.01499462 -0.2723291  1.59914271 -0.047870149
## 449 -0.48724019  1.01499462 -0.2723291  1.36613839 -0.141804663
## 450 -0.48724019  1.01499462 -0.2723291  1.36613839  0.188389387
## 451 -0.48724019  1.01499462 -0.2723291  1.36613839  0.660908458
## 452 -0.48724019  1.01499462 -0.2723291  1.36613839  0.527122938
## 453 -0.48724019  1.01499462 -0.2723291  1.36613839  0.017599361
## 454 -0.48724019  1.01499462 -0.2723291  1.36613839  1.577481596
## 455 -0.48724019  1.01499462 -0.2723291  1.36613839  0.631020203
## 456 -0.48724019  1.01499462 -0.2723291  1.36613839  0.342100410
## 457 -0.48724019  1.01499462 -0.2723291  1.36613839 -0.439263958
## 458 -0.48724019  1.01499462 -0.2723291  1.36613839 -0.496193967
## 459 -0.48724019  1.01499462 -0.2723291  1.36613839  0.023292362
## 460 -0.48724019  1.01499462 -0.2723291  1.36613839 -0.289822685
## 461 -0.48724019  1.01499462 -0.2723291  1.36613839  0.592592447
## 462 -0.48724019  1.01499462 -0.2723291  1.36613839  0.130036128
## 463 -0.48724019  1.01499462 -0.2723291  1.36613839  0.046064365
## 464 -0.48724019  1.01499462 -0.2723291  1.36613839  0.325021407
## 465 -0.48724019  1.01499462 -0.2723291  0.86561057 -0.107646658
## 466 -0.48724019  1.01499462 -0.2723291  0.86561057 -0.748109254
## 467 -0.48724019  1.01499462 -0.2723291  0.86561057 -0.473421963
## 468 -0.48724019  1.01499462 -0.2723291  0.25289548 -0.400836202
## 469 -0.48724019  1.01499462 -0.2723291  0.21837632 -0.510426469
## 470 -0.48724019  1.01499462 -0.2723291  0.21837632 -0.813578764
## 471 -0.48724019  1.01499462 -0.2723291  0.21837632 -0.167423167
## 472 -0.48724019  1.01499462 -0.2723291 -0.19585359 -0.079181654
## 473 -0.48724019  1.01499462 -0.2723291  0.21837632  0.216854391
## 474 -0.48724019  1.01499462 -0.2723291  0.51178918  0.989679257
## 475 -0.48724019  1.01499462 -0.2723291  0.25289548 -1.220628326
## 476 -0.48724019  1.01499462 -0.2723291  0.25289548 -0.174539418
## 477 -0.48724019  1.01499462 -0.2723291  0.51178918  0.283747151
## 478 -0.48724019  1.01499462 -0.2723291  0.51178918 -1.395688102
## 479 -0.48724019  1.01499462 -0.2723291  0.51178918 -0.141804663
## 480 -0.48724019  1.01499462 -0.2723291  0.51178918 -0.079181654
## 481 -0.48724019  1.01499462 -0.2723291 -0.19585359 -0.060679401
## 482 -0.48724019  1.01499462 -0.2723291 -0.19585359  0.662331708
## 483 -0.48724019  1.01499462 -0.2723291 -0.19585359  1.104962525
## 484 -0.48724019  1.01499462 -0.2723291 -0.19585359 -0.743839504
## 485 -0.48724019  1.01499462 -0.2723291  0.24426569 -0.588705230
## 486 -0.48724019  1.01499462 -0.2723291  0.24426569  0.038948114
## 487 -0.48724019  1.01499462 -0.2723291  0.24426569 -0.242855428
## 488 -0.48724019  1.01499462 -0.2723291  0.24426569 -0.540314723
## 489 -0.48724019  2.42017014 -0.2723291  0.46864023 -1.182200570
## 490 -0.48724019  2.42017014 -0.2723291  0.46864023 -1.239130578
## 491 -0.48724019  2.42017014 -0.2723291  0.46864023 -1.695993897
## 492 -0.48724019  2.42017014 -0.2723291  0.46864023 -0.429301206
## 493 -0.48724019  2.42017014 -0.2723291  0.46864023 -0.429301206
## 494 -0.48724019 -0.21088983 -0.2723291  0.26152527 -0.822118266
## 495 -0.48724019 -0.21088983 -0.2723291  0.26152527 -0.510426469
## 496 -0.48724019 -0.21088983 -0.2723291  0.26152527 -0.874778524
## 497 -0.48724019 -0.21088983 -0.2723291  0.26152527 -1.273288584
## 498 -0.48724019 -0.21088983 -0.2723291  0.26152527 -0.698295497
## 499 -0.48724019 -0.21088983 -0.2723291  0.26152527 -0.378064199
## 500 -0.48724019 -0.21088983 -0.2723291  0.26152527 -1.018526795
## 501 -0.48724019 -0.21088983 -0.2723291  0.26152527 -0.366678197
## 502 -0.48724019  0.11562398 -0.2723291  0.15796779  0.438881424
## 503 -0.48724019  0.11562398 -0.2723291  0.15796779 -0.234315927
## 504 -0.48724019  0.11562398 -0.2723291  0.15796779  0.983986256
## 505 -0.48724019  0.11562398 -0.2723291  0.15796779  0.724954717
## 506 -0.48724019  0.11562398 -0.2723291  0.15796779 -0.362408446
##              age           dis        rad         tax     ptratio
## 1   -0.119894767  0.1400749840 -0.9818712 -0.66594918 -1.45755797
## 2    0.366803426  0.5566090496 -0.8670245 -0.98635338 -0.30279450
## 3   -0.265548971  0.5566090496 -0.8670245 -0.98635338 -0.30279450
## 4   -0.809087830  1.0766711351 -0.7521778 -1.10502160  0.11292035
## 5   -0.510674339  1.0766711351 -0.7521778 -1.10502160  0.11292035
## 6   -0.350809969  1.0766711351 -0.7521778 -1.10502160  0.11292035
## 7   -0.070159185  0.8384142195 -0.5224844 -0.57694801 -1.50374851
## 8    0.977840575  1.0236248974 -0.5224844 -0.57694801 -1.50374851
## 9    1.116389695  1.0861216287 -0.5224844 -0.57694801 -1.50374851
## 10   0.615481336  1.3283202075 -0.5224844 -0.57694801 -1.50374851
## 11   0.913894827  1.2117799501 -0.5224844 -0.57694801 -1.50374851
## 12   0.508905089  1.1547920492 -0.5224844 -0.57694801 -1.50374851
## 13  -1.050660656  0.7863652700 -0.5224844 -0.57694801 -1.50374851
## 14  -0.240681180  0.4333252240 -0.6373311 -0.60068166  1.17530274
## 15   0.565745754  0.3166899868 -0.6373311 -0.60068166  1.17530274
## 16  -0.428965883  0.3341187865 -0.6373311 -0.60068166  1.17530274
## 17  -1.395257187  0.3341187865 -0.6373311 -0.60068166  1.17530274
## 18   0.466274590  0.2198105553 -0.6373311 -0.60068166  1.17530274
## 19  -1.135921653  0.0006920764 -0.6373311 -0.60068166  1.17530274
## 20   0.032864520  0.0006920764 -0.6373311 -0.60068166  1.17530274
## 21   1.048891406  0.0013569352 -0.6373311 -0.60068166  1.17530274
## 22   0.732715207  0.1031753182 -0.6373311 -0.60068166  1.17530274
## 23   0.821528746  0.0863638874 -0.6373311 -0.60068166  1.17530274
## 24   1.116389695  0.1425444597 -0.6373311 -0.60068166  1.17530274
## 25   0.906789743  0.2871037683 -0.6373311 -0.60068166  1.17530274
## 26   0.608376252  0.3132232229 -0.6373311 -0.60068166  1.17530274
## 27   0.771793164  0.4212152950 -0.6373311 -0.60068166  1.17530274
## 28   0.718505041  0.3126533438 -0.6373311 -0.60068166  1.17530274
## 29   0.917447368  0.3132707128 -0.6373311 -0.60068166  1.17530274
## 30   0.665216917  0.2108349609 -0.6373311 -0.60068166  1.17530274
## 31   0.906789743  0.2079855659 -0.6373311 -0.60068166  1.17530274
## 32   1.116389695  0.1804414138 -0.6373311 -0.60068166  1.17530274
## 33   0.476932215  0.0925850666 -0.6373311 -0.60068166  1.17530274
## 34   0.938762618 -0.0037244859 -0.6373311 -0.60068166  1.17530274
## 35   1.006260907 -0.0167367233 -0.6373311 -0.60068166  1.17530274
## 36  -0.013318520 -0.2064589434 -0.5224844 -0.76681717  0.34387304
## 37  -0.254891346 -0.1981007179 -0.5224844 -0.76681717  0.34387304
## 38  -0.961847117  0.0660856927 -0.5224844 -0.76681717  0.34387304
## 39  -1.363284313  0.0248169544 -0.5224844 -0.76681717  0.34387304
## 40  -1.661697804  0.7627152911 -0.7521778 -0.92701927 -0.07184181
## 41  -1.874850298  0.7627152911 -0.7521778 -0.92701927 -0.07184181
## 42  -2.333128159  0.9145880470 -0.7521778 -1.03975408 -0.25660396
## 43  -2.201684121  0.9145880470 -0.7521778 -1.03975408 -0.25660396
## 44  -2.205236663  0.9145880470 -0.7521778 -1.03975408 -0.25660396
## 45  -1.015135240  0.9145880470 -0.7521778 -1.03975408 -0.25660396
## 46  -1.235392817  0.6199131095 -0.7521778 -1.03975408 -0.25660396
## 47  -1.253155525  0.6199131095 -0.7521778 -1.03975408 -0.25660396
## 48   0.601271169  0.8996287230 -0.7521778 -1.03975408 -0.25660396
## 49   0.949420242  0.9853955139 -0.7521778 -1.03975408 -0.25660396
## 50  -0.233576097  1.0887810641 -0.7521778 -1.03975408 -0.25660396
## 51  -0.812640371  1.4340327636 -0.6373311 -0.98041997 -0.76469989
## 52  -0.198050682  1.4340327636 -0.6373311 -0.98041997 -0.76469989
## 53  -1.686565595  1.4340327636 -0.6373311 -0.98041997 -0.76469989
## 54  -1.675907970  1.4340327636 -0.6373311 -0.98041997 -0.76469989
## 55  -0.745142082  1.6738568465 -0.7521778  0.36053095  1.22149328
## 56  -1.658145262  2.3277455193 -0.5224844 -1.08128796 -0.25660396
## 57  -1.167894527  2.5609210138 -0.8670245 -0.56508119 -0.53374719
## 58  -0.997372532  2.1511780064 -0.5224844 -0.90328562 -1.54993904
## 59  -1.398809729  1.9089794276 -0.1779443 -0.73715011  0.57482574
## 60  -0.759352248  1.4897384367 -0.1779443 -0.73715011  0.57482574
## 61  -0.084369352  1.6290738544 -0.1779443 -0.73715011  0.57482574
## 62   0.881921953  1.4358373805 -0.1779443 -0.73715011  0.57482574
## 63  -0.027528687  1.6291213443 -0.1779443 -0.73715011  0.57482574
## 64  -0.894348827  1.9878601804 -0.1779443 -0.73715011  0.57482574
## 65  -0.322389636  2.5776849546 -0.7521778 -1.14062207  0.06672981
## 66  -1.803799466  1.3375332514 -0.6373311 -0.42267932 -1.08803366
## 67  -1.331311439  1.3375332514 -0.6373311 -0.42267932 -1.08803366
## 68  -1.675907970  1.2836321952 -0.6373311 -0.37521203  0.20530143
## 69  -1.128816570  1.2836321952 -0.6373311 -0.37521203  0.20530143
## 70  -1.263813149  1.2836321952 -0.6373311 -0.37521203  0.20530143
## 71  -2.201684121  0.7086717651 -0.6373311 -0.61254848  0.34387304
## 72  -1.814457091  0.7086717651 -0.6373311 -0.61254848  0.34387304
## 73  -2.159053622  0.7086717651 -0.6373311 -0.61254848  0.34387304
## 74  -2.215894287  0.7086717651 -0.6373311 -0.61254848  0.34387304
## 75  -2.222999370  0.2167712006 -0.5224844 -0.06074124  0.11292035
## 76  -0.837508162  0.3360183832 -0.5224844 -0.06074124  0.11292035
## 77   0.210491598  0.1221237952 -0.5224844 -0.06074124  0.11292035
## 78  -0.809087830  0.1403124336 -0.5224844 -0.06074124  0.11292035
## 79  -0.528437047  0.5789293108 -0.5224844 -0.06074124  0.11292035
## 80  -1.135921653  0.3360183832 -0.5224844 -0.06074124  0.11292035
## 81  -1.246050442  0.7625253315 -0.6373311 -0.75495035  0.25149196
## 82   0.064837394  0.7625253315 -0.6373311 -0.75495035  0.25149196
## 83  -1.292233482  0.7625253315 -0.6373311 -0.75495035  0.25149196
## 84  -0.777114956  0.7625253315 -0.6373311 -0.75495035  0.25149196
## 85  -0.730931915  0.4674704746 -0.7521778 -0.95668632  0.02053927
## 86  -0.443176049  0.3051974268 -0.7521778 -0.95668632  0.02053927
## 87  -0.833955621  0.3002109855 -0.7521778 -0.95668632  0.02053927
## 88  -0.418308258 -0.0225304932 -0.7521778 -0.95668632  0.02053927
## 89   0.629691502 -0.1773001341 -0.8670245 -0.82021787 -0.30279450
## 90  -0.194498140 -0.1807194081 -0.8670245 -0.82021787 -0.30279450
## 91  -0.087921893 -0.3337319220 -0.8670245 -0.82021787 -0.30279450
## 92   0.189176348 -0.3338269018 -0.8670245 -0.82021787 -0.30279450
## 93  -0.531989588 -0.0613297558 -0.6373311 -0.82021787 -0.11803234
## 94  -1.409467353 -0.0613297558 -0.6373311 -0.82021787 -0.11803234
## 95   0.309962761 -0.0855021237 -0.6373311 -0.82021787 -0.11803234
## 96  -0.382782843 -0.1423950448 -0.8670245 -0.78461740 -0.21041342
## 97   0.036417061 -0.1423950448 -0.8670245 -0.78461740 -0.21041342
## 98   0.263779721 -0.1423950448 -0.8670245 -0.78461740 -0.21041342
## 99  -1.125264029 -0.1423950448 -0.8670245 -0.78461740 -0.21041342
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## 317  0.519562713  0.0966691995 -0.6373311 -0.61848189 -0.02565127
## 318  0.111020434  0.1123883621 -0.6373311 -0.61848189 -0.02565127
## 319 -0.048843936 -0.1246813056 -0.6373311 -0.61848189 -0.02565127
## 320 -0.347257427  0.0982363667 -0.6373311 -0.61848189 -0.02565127
## 321 -0.578172628  0.3539695720 -0.5224844 -0.71934988  0.52863520
## 322 -0.507121797  0.3539695720 -0.5224844 -0.71934988  0.52863520
## 323 -0.663433626  0.4397838527 -0.5224844 -0.71934988  0.52863520
## 324  0.203386515  0.4397838527 -0.5224844 -0.71934988  0.52863520
## 325 -1.011582699  0.4397838527 -0.5224844 -0.71934988  0.52863520
## 326 -1.913928255  0.7697437989 -0.5224844 -0.71934988  0.52863520
## 327 -1.409467353  0.7697437989 -0.5224844 -0.71934988  0.52863520
## 328 -0.883691203  0.7697437989 -0.5224844 -0.71934988  0.52863520
## 329 -1.519596142  0.6741465952 -0.6373311  0.12912791 -0.71850935
## 330 -1.825114716  0.6741465952 -0.6373311  0.12912791 -0.71850935
## 331 -1.292233482  0.9871051509 -0.6373311  0.12912791 -0.71850935
## 332 -1.427230061  1.3514003073 -0.9818712 -0.61848189 -0.71850935
## 333 -1.608409681  1.3514003073 -0.9818712 -0.61848189 -0.71850935
## 334 -1.082633530  1.2648261879 -0.5224844 -1.09315478  0.80577843
## 335 -1.068423364  1.2648261879 -0.5224844 -1.09315478  0.80577843
## 336 -1.210525026  1.0401513886 -0.5224844 -1.09315478  0.80577843
## 337 -0.791325122  0.6819824315 -0.5224844 -1.09315478  0.80577843
## 338 -0.318837095  0.8642962245 -0.5224844 -1.09315478  0.80577843
## 339 -1.111053862  0.4830471675 -0.5224844 -1.09315478  0.80577843
## 340 -0.823297996  0.4830471675 -0.5224844 -1.09315478  0.80577843
## 341 -0.357915052  0.4830471675 -0.5224844 -1.09315478  0.80577843
## 342 -0.684748875  1.5400302593 -0.9818712 -0.73715011 -1.36517689
## 343 -0.315284553  1.1738829960 -0.9818712  0.08166062 -1.18041474
## 344 -0.432518424  0.9199069177 -0.5224844 -0.22687676 -0.39517558
## 345 -1.437887686  1.2681504821 -0.5224844 -0.22687676 -0.39517558
## 346 -0.713169208  2.0033893834 -0.7521778 -0.33367816  0.15911089
## 347 -0.578172628  2.0033893834 -0.7521778 -0.33367816  0.15911089
## 348 -1.452097852  2.2511442825 -0.6373311 -0.33961157 -0.25660396
## 349 -1.381047021  2.1602960705 -0.6373311 -0.76088376 -0.67231881
## 350 -1.210525026  2.3730983904 -0.9818712 -0.43454615  0.57482574
## 351 -0.858823412  2.3730983904 -0.9818712 -0.43454615  0.57482574
## 352 -1.160789444  3.2840499862 -0.6373311  0.01639310 -0.07184181
## 353 -1.778931675  3.2840499862 -0.6373311  0.01639310 -0.07184181
## 354 -1.153684361  3.9566021965 -0.5224844 -1.31269099 -0.67231881
## 355 -1.658145262  3.2248775491 -0.6373311 -0.44047956  1.63720813
## 356 -1.743406260  3.2248775491 -0.6373311 -0.44047956  1.63720813
## 357  1.024023615 -0.7944316108  1.6596029  1.52941294  0.80577843
## 358  0.796660955 -0.6125452271  1.6596029  1.52941294  0.80577843
## 359  0.526667797 -0.5092546567  1.6596029  1.52941294  0.80577843
## 360  0.452064424 -0.6106931203  1.6596029  1.52941294  0.80577843
## 361  0.690084708 -0.6063715378  1.6596029  1.52941294  0.80577843
## 362  0.800213497 -0.7121315839  1.6596029  1.52941294  0.80577843
## 363  0.981393116 -0.8032647354  1.6596029  1.52941294  0.80577843
## 364  0.725610124 -0.8977221812  1.6596029  1.52941294  0.80577843
## 365  0.508905089 -0.8977221812  1.6596029  1.52941294  0.80577843
## 366  0.686532167 -1.0361552904  1.6596029  1.52941294  0.80577843
## 367  0.810871121 -0.9700968153  1.6596029  1.52941294  0.80577843
## 368  1.116389695 -1.0848799457  1.6596029  1.52941294  0.80577843
## 369  1.116389695 -1.1694594886  1.6596029  1.52941294  0.80577843
## 370  1.002708366 -1.1579669285  1.6596029  1.52941294  0.80577843
## 371  1.027576156 -1.2312438711  1.6596029  1.52941294  0.80577843
## 372  1.116389695 -1.2470580136  1.6596029  1.52941294  0.80577843
## 373  0.746925373 -1.2658165310  1.6596029  1.52941294  0.80577843
## 374  1.116389695 -1.2446360278  1.6596029  1.52941294  0.80577843
## 375  1.116389695 -1.2623022771  1.6596029  1.52941294  0.80577843
## 376  1.041786323 -1.1771528552  1.6596029  1.52941294  0.80577843
## 377  0.878369411 -1.1635707388  1.6596029  1.52941294  0.80577843
## 378  1.073759197 -1.1573495596  1.6596029  1.52941294  0.80577843
## 379  0.981393116 -1.1440048928  1.6596029  1.52941294  0.80577843
## 380  1.116389695 -1.1440048928  1.6596029  1.52941294  0.80577843
## 381  0.828633829 -1.1295679579  1.6596029  1.52941294  0.80577843
## 382  1.084416821 -1.0807958128  1.6596029  1.52941294  0.80577843
## 383  1.116389695 -1.0517319833  1.6596029  1.52941294  0.80577843
## 384  1.116389695 -1.0741947142  1.6596029  1.52941294  0.80577843
## 385  0.803766038 -1.1186452769  1.6596029  1.52941294  0.80577843
## 386  1.048891406 -1.1250089259  1.6596029  1.52941294  0.80577843
## 387  1.116389695 -1.1054905698  1.6596029  1.52941294  0.80577843
## 388  0.743372832 -1.0811757321  1.6596029  1.52941294  0.80577843
## 389  1.116389695 -1.0474104008  1.6596029  1.52941294  0.80577843
## 390  1.077311738 -0.9815893753  1.6596029  1.52941294  0.80577843
## 391  1.009813449 -0.8873693792  1.6596029  1.52941294  0.80577843
## 392  0.494694923 -0.7727762084  1.6596029  1.52941294  0.80577843
## 393  1.009813449 -0.9616910999  1.6596029  1.52941294  0.80577843
## 394  0.853501620 -0.9516232374  1.6596029  1.52941294  0.80577843
## 395  0.928104993 -0.9559448199  1.6596029  1.52941294  0.80577843
## 396  1.073759197 -0.9827291333  1.6596029  1.52941294  0.80577843
## 397  0.974288033 -1.0059517029  1.6596029  1.52941294  0.80577843
## 398  1.077311738 -1.0265623271  1.6596029  1.52941294  0.80577843
## 399  1.116389695 -1.0948528283  1.6596029  1.52941294  0.80577843
## 400  0.327725469 -1.0897239172  1.6596029  1.52941294  0.80577843
## 401  1.116389695 -1.0477428302  1.6596029  1.52941294  0.80577843
## 402  1.116389695 -1.0547238481  1.6596029  1.52941294  0.80577843
## 403  1.116389695 -1.0239028917  1.6596029  1.52941294  0.80577843
## 404  0.974288033 -0.9936043244  1.6596029  1.52941294  0.80577843
## 405  0.597718628 -1.0389097056  1.6596029  1.52941294  0.80577843
## 406  1.116389695 -1.1253413553  1.6596029  1.52941294  0.80577843
## 407  1.116389695 -1.2427839210  1.6596029  1.52941294  0.80577843
## 408  1.116389695 -1.1919222195  1.6596029  1.52941294  0.80577843
## 409  1.041786323 -1.1114268095  1.6596029  1.52941294  0.80577843
## 410  1.116389695 -1.1062978984  1.6596029  1.52941294  0.80577843
## 411  1.116389695 -1.1312301050  1.6596029  1.52941294  0.80577843
## 412  1.116389695 -1.0768541496  1.6596029  1.52941294  0.80577843
## 413  1.116389695 -1.0643168114  1.6596029  1.52941294  0.80577843
## 414  1.116389695 -1.0474578907  1.6596029  1.52941294  0.80577843
## 415  1.116389695 -1.0147848276  1.6596029  1.52941294  0.80577843
## 416  1.116389695 -0.9309651233  1.6596029  1.52941294  0.80577843
## 417  0.789555872 -0.9381835908  1.6596029  1.52941294  0.80577843
## 418  0.729162665 -1.0198662487  1.6596029  1.52941294  0.80577843
## 419  1.116389695 -0.9462093868  1.6596029  1.52941294  0.80577843
## 420  0.281542429 -0.9502935197  1.6596029  1.52941294  0.80577843
## 421  1.116389695 -0.9194725633  1.6596029  1.52941294  0.80577843
## 422  0.949420242 -0.9120166463  1.6596029  1.52941294  0.80577843
## 423  0.675874542 -0.8756393696  1.6596029  1.52941294  0.80577843
## 424  0.587061003 -0.8421114879  1.6596029  1.52941294  0.80577843
## 425  0.071942477 -0.8223081923  1.6596029  1.52941294  0.80577843
## 426  0.952972784 -0.8953951752  1.6596029  1.52941294  0.80577843
## 427 -0.315284553 -0.8536040479  1.6596029  1.52941294  0.80577843
## 428  0.359698343 -0.9175729666  1.6596029  1.52941294  0.80577843
## 429  0.338383094 -0.8830477967  1.6596029  1.52941294  0.80577843
## 430  0.960077867 -0.8675660836  1.6596029  1.52941294  0.80577843
## 431  0.622586419 -0.8274371034  1.6596029  1.52941294  0.80577843
## 432  0.913894827 -0.8105781827  1.6596029  1.52941294  0.80577843
## 433  0.221149222 -0.7572944954  1.6596029  1.52941294  0.80577843
## 434  0.686532167 -0.7024911307  1.6596029  1.52941294  0.80577843
## 435  0.938762618 -0.7469416934  1.6596029  1.52941294  0.80577843
## 436  0.924552451 -0.7932443629  1.6596029  1.52941294  0.80577843
## 437  0.878369411 -0.8512295520  1.6596029  1.52941294  0.80577843
## 438  1.116389695 -0.8932106390  1.6596029  1.52941294  0.80577843
## 439  0.686532167 -0.9376612017  1.6596029  1.52941294  0.80577843
## 440  0.899684660 -0.9392758589  1.6596029  1.52941294  0.80577843
## 441  0.846396537 -0.9160057994  1.6596029  1.52941294  0.80577843
## 442  1.016918532 -0.8215483536  1.6596029  1.52941294  0.80577843
## 443  1.116389695 -0.8501847738  1.6596029  1.52941294  0.80577843
## 444  1.116389695 -0.8627221120  1.6596029  1.52941294  0.80577843
## 445  0.995603282 -0.9020437636  1.6596029  1.52941294  0.80577843
## 446  0.931657534 -0.8582105699  1.6596029  1.52941294  0.80577843
## 447  0.988498199 -0.8182715493  1.6596029  1.52941294  0.80577843
## 448  0.995603282 -0.7584342534  1.6596029  1.52941294  0.80577843
## 449  1.070206655 -0.7282306659  1.6596029  1.52941294  0.80577843
## 450  1.055996489 -0.7646079427  1.6596029  1.52941294  0.80577843
## 451  0.853501620 -0.6987869171  1.6596029  1.52941294  0.80577843
## 452  1.052443947 -0.6837801032  1.6596029  1.52941294  0.80577843
## 453  0.825081288 -0.6776064140  1.6596029  1.52941294  0.80577843
## 454  1.091521905 -0.6374774338  1.6596029  1.52941294  0.80577843
## 455  0.906789743 -0.6168668096  1.6596029  1.52941294  0.80577843
## 456  0.636796585 -0.6455032298  1.6596029  1.52941294  0.80577843
## 457  0.686532167 -0.5767378294  1.6596029  1.52941294  0.80577843
## 458  0.416539008 -0.4824228534  1.6596029  1.52941294  0.80577843
## 459  0.537325421 -0.4805707466  1.6596029  1.52941294  0.80577843
## 460  0.562193212 -0.5117241325  1.6596029  1.52941294  0.80577843
## 461  0.761135540 -0.5687120333  1.6596029  1.52941294  0.80577843
## 462  0.704294875 -0.5831489682  1.6596029  1.52941294  0.80577843
## 463  0.512457630 -0.5036983364  1.6596029  1.52941294  0.80577843
## 464  0.757582998 -0.4717851119  1.6596029  1.52941294  0.80577843
## 465 -0.112789684 -0.3949464255  1.6596029  1.52941294  0.80577843
## 466 -0.723826832 -0.3459843207  1.6596029  1.52941294  0.80577843
## 467  0.572850837 -0.4385896596  1.6596029  1.52941294  0.80577843
## 468  0.920999910 -0.5958762661  1.6596029  1.52941294  0.80577843
## 469  0.086152643 -0.4210658801  1.6596029  1.52941294  0.80577843
## 470 -0.421860800 -0.4612898402  1.6596029  1.52941294  0.80577843
## 471  0.547983046 -0.3617034834  1.6596029  1.52941294  0.80577843
## 472  0.786003330 -0.3304076278  1.6596029  1.52941294  0.80577843
## 473  0.228254306 -0.4267171803  1.6596029  1.52941294  0.80577843
## 474 -0.034633770 -0.5993905200  1.6596029  1.52941294  0.80577843
## 475  0.952972784 -0.6483526248  1.6596029  1.52941294  0.80577843
## 476  1.024023615 -0.7546350600  1.6596029  1.52941294  0.80577843
## 477  0.889027036 -0.7074775720  1.6596029  1.52941294  0.80577843
## 478  1.020471073 -0.8046419430  1.6596029  1.52941294  0.80577843
## 479  0.999155824 -0.7714939807  1.6596029  1.52941294  0.80577843
## 480  0.690084708 -0.8756393696  1.6596029  1.52941294  0.80577843
## 481 -0.137657475 -0.1761128861  1.6596029  1.52941294  0.80577843
## 482  0.224701764 -0.2200410597  1.6596029  1.52941294  0.80577843
## 483  0.299305137 -0.1825715149  1.6596029  1.52941294  0.80577843
## 484 -1.004477616  0.1440166471  1.6596029  1.52941294  0.80577843
## 485 -0.947636951 -0.0337381137  1.6596029  1.52941294  0.80577843
## 486 -0.592382795  0.0933923952  1.6596029  1.52941294  0.80577843
## 487  0.398776300 -0.1183176566  1.6596029  1.52941294  0.80577843
## 488 -0.546199754 -0.3052379716  1.6596029  1.52941294  0.80577843
## 489  0.857054162 -0.9375187319 -0.6373311  1.79641644  0.75958789
## 490  1.055996489 -0.9686246278 -0.6373311  1.79641644  0.75958789
## 491  1.045338864 -0.9367114033 -0.6373311  1.79641644  0.75958789
## 492  1.073759197 -0.9151034909 -0.6373311  1.79641644  0.75958789
## 493  0.530220338 -0.8002728706 -0.6373311  1.79641644  0.75958789
## 494 -0.517779422 -0.6711952751 -0.4076377 -0.10227512  0.34387304
## 495 -0.922769160 -0.6711952751 -0.4076377 -0.10227512  0.34387304
## 496 -1.413019895 -0.4732098094 -0.4076377 -0.10227512  0.34387304
## 497  0.153650933 -0.4732098094 -0.4076377 -0.10227512  0.34387304
## 498  0.071942477 -0.4285217972 -0.4076377 -0.10227512  0.34387304
## 499 -0.116342226 -0.6581830377 -0.4076377 -0.10227512  0.34387304
## 500  0.174966182 -0.6625521101 -0.4076377 -0.10227512  0.34387304
## 501  0.395223759 -0.6158695213 -0.4076377 -0.10227512  0.34387304
## 502  0.018654354 -0.6251775451 -0.9818712 -0.80241764  1.17530274
## 503  0.288647512 -0.7159307773 -0.9818712 -0.80241764  1.17530274
## 504  0.796660955 -0.7729186782 -0.9818712 -0.80241764  1.17530274
## 505  0.736267749 -0.6677760011 -0.9818712 -0.80241764  1.17530274
## 506  0.434301716 -0.6126402069 -0.9818712 -0.80241764  1.17530274
##            black        lstat         medv
## 1    0.440615895 -1.074498970  0.159527789
## 2    0.440615895 -0.491952525 -0.101423917
## 3    0.396035074 -1.207532413  1.322937477
## 4    0.415751408 -1.360170785  1.181588636
## 5    0.440615895 -1.025486649  1.486032293
## 6    0.410165113 -1.042290874  0.670558212
## 7    0.426376321 -0.031236706  0.039924923
## 8    0.440615895  0.909799859  0.496590409
## 9    0.328123258  2.419379350 -0.655946292
## 10   0.328999540  0.622727693 -0.394994586
## 11   0.392639483  1.091845624 -0.819041108
## 12   0.440615895  0.086392864 -0.394994586
## 13   0.370513376  0.428078760 -0.090550930
## 14   0.440615895 -0.615183504 -0.231899770
## 15   0.255720500 -0.335113097 -0.471105500
## 16   0.426595391 -0.585776111 -0.286264709
## 17   0.330533033 -0.850442645  0.061670899
## 18   0.329437681  0.282442149 -0.547216415
## 19  -0.741378303 -0.134862757 -0.253645746
## 20   0.375442459 -0.192277190 -0.471105500
## 21   0.217930861  1.171665689 -0.971262937
## 22   0.392749018  0.164812578 -0.318883672
## 23   0.440615895  0.849584722 -0.797295133
## 24   0.414765591  1.012025558 -0.873406047
## 25   0.412465352  0.510699530 -0.753803182
## 26  -0.583319029  0.540106923 -0.938643973
## 27   0.221326452  0.302047077 -0.645073304
## 28  -0.550896614  0.647934029 -0.840787084
## 29   0.342472368  0.020576319 -0.449359525
## 30   0.258020739 -0.094252548 -0.166661844
## 31   0.038293155  1.392921311 -1.069119826
## 32   0.219683424  0.054184768 -0.873406047
## 33  -1.359047220  2.108501199 -1.014754888
## 34   0.022958229  0.797771697 -1.025627875
## 35  -1.186967442  1.076441751 -0.982135924
## 36   0.440615895 -0.416333515 -0.394994586
## 37   0.228774844 -0.174072614 -0.275391721
## 38   0.440615895 -0.543765550 -0.166661844
## 39   0.402607185 -0.353317674  0.235638703
## 40   0.426704926 -1.166922204  0.898890955
## 41   0.426595391 -1.494604580  1.344683452
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## 374  0.440615895  3.097149734 -0.949516961
## 375  0.440615895  3.545262384 -0.949516961
## 376  0.440615895  0.110198849 -0.819041108
## 377  0.069510683  1.482543841 -0.938643973
## 378  0.440615895  1.202473434 -1.003881900
## 379  0.440615895  1.545559682 -1.025627875
## 380  0.406002776  1.278092444 -1.340944520
## 381  0.440615895  0.638131565 -1.319198544
## 382  0.440615895  1.180067802 -1.264833606
## 383  0.440615895  1.532956514 -1.221341655
## 384  0.440615895  1.667390309 -1.112611777
## 385 -0.775991422  2.517403992 -1.493166348
## 386  0.440615895  2.542610329 -1.667134152
## 387  0.440615895  2.188321265 -1.308325557
## 388  0.440615895  2.707851869 -1.645388177
## 389  0.177950518  2.516003640 -1.340944520
## 390  0.440615895  1.147859705 -1.199595679
## 391  0.413560704  0.624128045 -0.808168120
## 392  0.237756730  0.855186130  0.072543887
## 393  0.440615895  1.824229736 -1.395309459
## 394  0.440615895  0.352459751 -0.949516961
## 395  0.440615895  0.517701290 -1.069119826
## 396  0.386724583  0.625528397 -1.025627875
## 397  0.440615895  0.940607604 -1.090865802
## 398  0.398992524  1.017626966 -1.525785311
## 399  0.440615895  2.511802584 -1.906339882
## 400 -0.202793791  2.424980758 -1.764991042
## 401  0.440615895  1.976868108 -1.841101956
## 402  0.440615895  1.073641047 -1.667134152
## 403  0.212892242  1.072240695 -1.134357753
## 404  0.440615895  0.996621685 -1.547531287
## 405 -0.298089403  2.062289582 -1.525785311
## 406  0.309940417  1.446134688 -1.906339882
## 407  0.148376017  1.496547361 -1.156103728
## 408 -0.269281649 -0.073247267  0.583574310
## 409 -0.460420549  1.925055083 -0.579835378
## 410 -1.942212553  0.998022037  0.540082359
## 411 -3.878356510 -0.356118378 -0.819041108
## 412 -3.522914830  1.199672730 -0.579835378
## 413 -3.591483857  3.041135653 -0.503724464
## 414 -1.595971828  1.040032598 -0.677692268
## 415 -2.939968567  3.406627533 -1.688880128
## 416 -3.608352275  2.296148371 -1.667134152
## 417 -3.670568261  1.839633609 -1.634515189
## 418 -2.511795523  1.958663532 -1.319198544
## 419 -3.726650277  1.115651608 -1.493166348
## 420 -3.376137680  1.412526239 -1.536658299
## 421 -0.415401588  0.331454470 -0.634200317
## 422 -0.401928760  0.426678408 -0.906025010
## 423 -0.713337295  0.202622083 -0.188407819
## 424 -3.879232792  1.489545601 -0.993008912
## 425 -3.866855315  0.631129805 -1.177849704
## 426 -3.822712635  1.643584324 -1.547531287
## 427 -3.636831424  0.425278056 -1.340944520
## 428 -3.700690438  0.261436868 -1.264833606
## 429 -2.847301799  1.241683291 -1.253960618
## 430 -3.241738006  1.600173411 -1.417055434
## 431 -2.992764527  0.698346703 -0.873406047
## 432 -3.015985986  0.985418869 -0.916897998
## 433 -2.833938506 -0.087250788 -0.699438243
## 434 -2.809402625  0.499496714 -0.895152022
## 435 -2.804583076  0.352459751 -1.177849704
## 436 -2.703591634  1.486744897 -0.993008912
## 437 -3.605723431  0.755761136 -1.406182446
## 438 -3.804748865  1.932056843 -1.504039336
## 439 -3.151590547  2.992123331 -1.536658299
## 440  0.440615895  1.432131167 -1.058246839
## 441  0.380919218  1.324304061 -1.308325557
## 442  0.320784401  0.961612885 -0.590708366
## 443  0.427362137  0.551309739 -0.449359525
## 444  0.329218610  0.867789298 -0.775549157
## 445 -1.272295352  1.559563202 -1.275706593
## 446 -3.435177145  1.586169891 -1.166976716
## 447 -0.423507192  0.719351983 -0.829914096
## 448  0.348825409  0.530304459 -1.079992814
## 449  0.440615895  0.766963952 -0.916897998
## 450 -0.574665749  0.932205492 -1.036500863
## 451 -3.903330533  0.670339662 -0.993008912
## 452 -0.015160016  0.710949871 -0.797295133
## 453  0.311254840  0.646533677 -0.699438243
## 454  0.210263398  0.572315020 -0.514597451
## 455 -3.833666154  0.848184370 -0.829914096
## 456 -3.349082489  0.766963952 -0.916897998
## 457 -3.792042783  0.890194931 -1.069119826
## 458 -3.868498343  0.600322060 -0.982135924
## 459 -0.925178346  0.500897066 -0.829914096
## 460  0.440615895  0.286643205 -0.275391721
## 461 -1.111169093  0.527503755 -0.666819280
## 462  0.380700148  0.279641445 -0.525470439
## 463  0.440615895  0.187218211 -0.329756660
## 464  0.406879058 -0.330912041 -0.253645746
## 465  0.440615895  0.079391104 -0.123169893
## 466 -0.243979021  0.206823139 -0.286264709
## 467 -3.665748713  0.629729453 -0.384121599
## 468 -0.278044464  1.213676250 -0.373248611
## 469  0.132164810  0.766963952 -0.373248611
## 470  0.440615895  0.295045317 -0.264518733
## 471  0.440615895  0.509299178 -0.286264709
## 472  0.423418871  0.030378783 -0.318883672
## 473  0.401949974  0.239031236  0.072543887
## 474  0.197228711 -0.139063813  0.790161078
## 475 -0.044844052  0.768364304 -0.949516961
## 476 -0.590548351  1.602974115 -1.003881900
## 477  0.433057967  0.843983314 -0.634200317
## 478 -0.078799960  1.716402630 -1.145230740
## 479  0.252215374  0.752960432 -0.862533059
## 480  0.291867112  0.063987232 -0.123169893
## 481  0.440615895 -0.267896200  0.050797911
## 482  0.398663919 -0.688001809  0.126908825
## 483  0.422871195 -0.790227508  0.268257666
## 484  0.397020891 -0.312707465 -0.079677942
## 485  0.153962312  0.096195329 -0.210153795
## 486  0.349920761 -0.290301832 -0.144915868
## 487  0.394392046  0.325853062 -0.373248611
## 488  0.345539353 -0.168471205 -0.210153795
## 489  0.420790026  0.757161488 -0.797295133
## 490 -0.138277566  1.584769539 -1.688880128
## 491 -0.418906714  2.384370549 -1.569277262
## 492  0.366241503  0.758561840 -0.971262937
## 493  0.440615895  0.097595681 -0.264518733
## 494  0.440615895 -0.090051492 -0.079677942
## 495  0.440615895  0.131204129  0.213892727
## 496  0.401073693  0.692745294  0.061670899
## 497  0.440615895  1.188469914 -0.308010684
## 498  0.440615895  0.202622083 -0.460232513
## 499  0.440615895  0.037380543 -0.144915868
## 500  0.428238419  0.342657286 -0.547216415
## 501  0.440615895  0.234830180 -0.623327329
## 502  0.386834118 -0.417733867 -0.014440015
## 503  0.440615895 -0.500354637 -0.210153795
## 504  0.440615895 -0.982075736  0.148654801
## 505  0.402826256 -0.864446166 -0.057931966
## 506  0.440615895 -0.668396881 -1.156103728
boston_scaled<-data.frame(boston_scaled, crime)

Divide the dataset into train and test

n<-nrow(boston_scaled)
ind<-sample(n, size = n*0.8)
train<-boston_scaled[ind, ]
test<-boston_scaled[-ind, ]
correct_classes<-test$crime

LInear Discriminant Analysis

lda.fit<-lda(crime~., data=train)
lda.fit
## Call:
## lda(crime ~ ., data = train)
## 
## Prior probabilities of groups:
##       low   med_low  med_high      high 
## 0.2549505 0.2301980 0.2648515 0.2500000 
## 
## Group means:
##                   zn      indus        chas        nox          rm
## low       0.99014775 -0.9159400 -0.15765625 -0.8709880  0.37229266
## med_low  -0.03933548 -0.3666704  0.02401181 -0.6223415 -0.06205674
## med_high -0.39026573  0.2348380  0.24280554  0.4341211  0.06118806
## high     -0.48724019  1.0171306 -0.03844192  1.0602508 -0.36424035
##                 age        dis        rad        tax     ptratio
## low      -0.8594787  0.8995356 -0.7053471 -0.7349035 -0.40459307
## med_low  -0.3681525  0.4149502 -0.5373033 -0.4786320 -0.05197491
## med_high  0.4395807 -0.4300579 -0.4205177 -0.2951387 -0.34423648
## high      0.8158306 -0.8436114  1.6379981  1.5139626  0.78062517
##                black        lstat        medv
## low       0.37902415 -0.747428399  0.42501870
## med_low   0.31084261 -0.223672179  0.04892729
## med_high  0.08986887  0.004361028  0.16582802
## high     -0.92053883  0.936378818 -0.69437854
## 
## Coefficients of linear discriminants:
##                 LD1         LD2         LD3
## zn       0.07508176  0.63518169 -1.12624756
## indus    0.01502145 -0.28482992  0.18073114
## chas    -0.07981630 -0.06239136  0.16457139
## nox      0.38896564 -0.75046323 -1.31343202
## rm      -0.10620673 -0.04914125 -0.05026209
## age      0.24657931 -0.26154525 -0.12107561
## dis     -0.08081170 -0.18204001  0.29354041
## rad      3.12450345  0.92605754 -0.19738488
## tax      0.05398304  0.02940260  0.73567524
## ptratio  0.10410133  0.03961344 -0.26805247
## black   -0.13273978  0.01771556  0.08506990
## lstat    0.23965345 -0.22106990  0.43743850
## medv     0.22524316 -0.45319558 -0.05130395
## 
## Proportion of trace:
##    LD1    LD2    LD3 
## 0.9462 0.0420 0.0118
lda.arrows<-function(x, myscale=1, arrow_heads= 0.1, color="red", tex=0.75, choices =c (1,2)){
  heads<-coef(x)
  arrows(x0 = 0, y0=0, x1 = myscale*heads[,choices[1]],y1=myscale*heads[,choices[2]], col=color, length= arrow_heads)
  text(myscale * heads[,choices], labels = row.names(heads), 
       cex = tex, col=color, pos=3)
}

Convert target classes to numeric data

classes<-as.numeric(train$crime)

Plot the lda results

plot(lda.fit, dimen=2, col=classes, pch=classes)
lda.arrows(lda.fit, myscale=1)

Predict the classes with the LDA model and Cross-tabulate the results

lda.pred<-predict(lda.fit, newdata = test)
table(correct=correct_classes, predict=lda.pred$class)
##           predict
## correct    low med_low med_high high
##   low       11      11        2    0
##   med_low    2      20       11    0
##   med_high   1       8        9    1
##   high       0       0        0   26

Distance measures

library(MASS)
data("Boston")
boston_scaled<-scale(Boston)
boston_scaled<-as.data.frame(boston_scaled)
Euclidean distane matrix
dist_eu<-dist(boston_scaled)
summary(dist_eu)
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##  0.1343  3.4625  4.8241  4.9111  6.1863 14.3970

K-means

km<-kmeans(boston_scaled, centers=3)
pairs(boston_scaled, col=km$cluster)

Investigate the optimal number of clusters

set.seed(11)
k_max<-20
twcss <- sapply(1:k_max, function(k){kmeans(boston_scaled, k)$tot.withinss})
qqplot(x=1:k_max, y = twcss, geom="Line")
## Warning in plot.window(...): "geom" is not a graphical parameter
## Warning in plot.xy(xy, type, ...): "geom" is not a graphical parameter
## Warning in axis(side = side, at = at, labels = labels, ...): "geom" is not
## a graphical parameter

## Warning in axis(side = side, at = at, labels = labels, ...): "geom" is not
## a graphical parameter
## Warning in box(...): "geom" is not a graphical parameter
## Warning in title(...): "geom" is not a graphical parameter

Perform k-means with different number of clusters

km<-kmeans(boston_scaled, centers=6)
pairs(boston_scaled, col = km$cluster)

library(GGally)
## Registered S3 method overwritten by 'GGally':
##   method from   
##   +.gg   ggplot2
## 
## Attaching package: 'GGally'
## The following object is masked from 'package:dplyr':
## 
##     nasa
ggpairs(boston_scaled)

Super Bonus

model_predictors<- dplyr::select(train, -crime)
Check th dimensions
dim(model_predictors)
## [1] 404  13
dim(lda.fit$scaling)
## [1] 13  3
Matrix multiplication
matrix_product <- as.matrix(model_predictors)%*%lda.fit$scaling
matrix_product <- as.data.frame(matrix_product)
Create a 3D plot
library(plotly)
## 
## Attaching package: 'plotly'
## The following object is masked from 'package:MASS':
## 
##     select
## The following object is masked from 'package:ggplot2':
## 
##     last_plot
## The following object is masked from 'package:stats':
## 
##     filter
## The following object is masked from 'package:graphics':
## 
##     layout
plot_ly(x=matrix_product$LD1, y=matrix_product$LD2, z=matrix_product$LD3, type="scatter3d", mode="markers")
plot_ly(x=matrix_product$LD1, y=matrix_product$LD2, z=matrix_product$LD3, type="scatter3d", mode="markers", color = train$crime)
km <-kmeans(boston_scaled, centers = 6)
plot_ly(x = matrix_product$LD1, y = matrix_product$LD2, z = matrix_product$LD3, type= 'scatter3d', mode='markers', color = km$cluster[ind])

Dimensionality Reduction Techniques (Excercise 5)

Human data

###1.Read and Take a look at data

human <- read.table("http://s3.amazonaws.com/assets.datacamp.com/production/course_2218/datasets/human2.txt", sep=",", header=T)
library(MASS);library(tidyr);library(dplyr);library(GGally);library(ggplot2);library(dplyr);library(tidyr);library(corrplot)
str(human);dim(human);summary(human)
## 'data.frame':    155 obs. of  8 variables:
##  $ Edu2.FM  : num  1.007 0.997 0.983 0.989 0.969 ...
##  $ Labo.FM  : num  0.891 0.819 0.825 0.884 0.829 ...
##  $ Edu.Exp  : num  17.5 20.2 15.8 18.7 17.9 16.5 18.6 16.5 15.9 19.2 ...
##  $ Life.Exp : num  81.6 82.4 83 80.2 81.6 80.9 80.9 79.1 82 81.8 ...
##  $ GNI      : int  64992 42261 56431 44025 45435 43919 39568 52947 42155 32689 ...
##  $ Mat.Mor  : int  4 6 6 5 6 7 9 28 11 8 ...
##  $ Ado.Birth: num  7.8 12.1 1.9 5.1 6.2 3.8 8.2 31 14.5 25.3 ...
##  $ Parli.F  : num  39.6 30.5 28.5 38 36.9 36.9 19.9 19.4 28.2 31.4 ...
## [1] 155   8
##     Edu2.FM          Labo.FM          Edu.Exp         Life.Exp    
##  Min.   :0.1717   Min.   :0.1857   Min.   : 5.40   Min.   :49.00  
##  1st Qu.:0.7264   1st Qu.:0.5984   1st Qu.:11.25   1st Qu.:66.30  
##  Median :0.9375   Median :0.7535   Median :13.50   Median :74.20  
##  Mean   :0.8529   Mean   :0.7074   Mean   :13.18   Mean   :71.65  
##  3rd Qu.:0.9968   3rd Qu.:0.8535   3rd Qu.:15.20   3rd Qu.:77.25  
##  Max.   :1.4967   Max.   :1.0380   Max.   :20.20   Max.   :83.50  
##       GNI            Mat.Mor         Ado.Birth         Parli.F     
##  Min.   :   581   Min.   :   1.0   Min.   :  0.60   Min.   : 0.00  
##  1st Qu.:  4198   1st Qu.:  11.5   1st Qu.: 12.65   1st Qu.:12.40  
##  Median : 12040   Median :  49.0   Median : 33.60   Median :19.30  
##  Mean   : 17628   Mean   : 149.1   Mean   : 47.16   Mean   :20.91  
##  3rd Qu.: 24512   3rd Qu.: 190.0   3rd Qu.: 71.95   3rd Qu.:27.95  
##  Max.   :123124   Max.   :1100.0   Max.   :204.80   Max.   :57.50

155 oberservations and 8 variables

To see structure and relationships

cor(human)
##                Edu2.FM      Labo.FM     Edu.Exp   Life.Exp         GNI
## Edu2.FM    1.000000000  0.009564039  0.59325156  0.5760299  0.43030485
## Labo.FM    0.009564039  1.000000000  0.04732183 -0.1400125 -0.02173971
## Edu.Exp    0.593251562  0.047321827  1.00000000  0.7894392  0.62433940
## Life.Exp   0.576029853 -0.140012504  0.78943917  1.0000000  0.62666411
## GNI        0.430304846 -0.021739705  0.62433940  0.6266641  1.00000000
## Mat.Mor   -0.660931770  0.240461075 -0.73570257 -0.8571684 -0.49516234
## Ado.Birth -0.529418415  0.120158862 -0.70356489 -0.7291774 -0.55656208
## Parli.F    0.078635285  0.250232608  0.20608156  0.1700863  0.08920818
##              Mat.Mor  Ado.Birth     Parli.F
## Edu2.FM   -0.6609318 -0.5294184  0.07863528
## Labo.FM    0.2404611  0.1201589  0.25023261
## Edu.Exp   -0.7357026 -0.7035649  0.20608156
## Life.Exp  -0.8571684 -0.7291774  0.17008631
## GNI       -0.4951623 -0.5565621  0.08920818
## Mat.Mor    1.0000000  0.7586615 -0.08944000
## Ado.Birth  0.7586615  1.0000000 -0.07087810
## Parli.F   -0.0894400 -0.0708781  1.00000000
ggpairs(human)

cor(human)%>%corrplot(method="circle", type="upper", title="Graph 1. Correlations between variables in the human data")

  • Hypothesis
      1. negative relationship between Edu2.FM and Mat.Mor
      1. negative relationship between Edu.Exp and Mat.Mor
      1. negative relationship between Life.Exp and Mat.Mor
      1. negative relationship between GNI and Mat.Mor

2.Principal component analysis

pca_human <- prcomp(human)
pca_human
## Standard deviations (1, .., p=8):
## [1] 1.854416e+04 1.855219e+02 2.518701e+01 1.145441e+01 3.766241e+00
## [6] 1.565912e+00 1.912052e-01 1.591112e-01
## 
## Rotation (n x k) = (8 x 8):
##                     PC1           PC2           PC3           PC4
## Edu2.FM   -5.607472e-06  0.0006713951 -3.412027e-05 -2.736326e-04
## Labo.FM    2.331945e-07 -0.0002819357  5.302884e-04 -4.692578e-03
## Edu.Exp   -9.562910e-05  0.0075529759  1.427664e-02 -3.313505e-02
## Life.Exp  -2.815823e-04  0.0283150248  1.294971e-02 -6.752684e-02
## GNI       -9.999832e-01 -0.0057723054 -5.156742e-04  4.932889e-05
## Mat.Mor    5.655734e-03 -0.9916320120  1.260302e-01 -6.100534e-03
## Ado.Birth  1.233961e-03 -0.1255502723 -9.918113e-01  5.301595e-03
## Parli.F   -5.526460e-05  0.0032317269 -7.398331e-03 -9.971232e-01
##                     PC5           PC6           PC7           PC8
## Edu2.FM   -0.0022935252  2.180183e-02  6.998623e-01  7.139410e-01
## Labo.FM    0.0022190154  3.264423e-02  7.132267e-01 -7.001533e-01
## Edu.Exp    0.1431180282  9.882477e-01 -3.826887e-02  7.776451e-03
## Life.Exp   0.9865644425 -1.453515e-01  5.380452e-03  2.281723e-03
## GNI       -0.0001135863 -2.711698e-05 -8.075191e-07 -1.176762e-06
## Mat.Mor    0.0266373214  1.695203e-03  1.355518e-04  8.371934e-04
## Ado.Birth  0.0188618600  1.273198e-02 -8.641234e-05 -1.707885e-04
## Parli.F   -0.0716401914 -2.309896e-02 -2.642548e-03  2.680113e-03
biplot(pca_human, choices = 1:2, cex = c(0.8, 1), col= c("grey40", "deeppink2"), main = "Grpagh 2. PCA with human data, unscaled")
## Warning in arrows(0, 0, y[, 1L] * 0.8, y[, 2L] * 0.8, col = col[2L], length
## = arrow.len): zero-length arrow is of indeterminate angle and so skipped

## Warning in arrows(0, 0, y[, 1L] * 0.8, y[, 2L] * 0.8, col = col[2L], length
## = arrow.len): zero-length arrow is of indeterminate angle and so skipped

## Warning in arrows(0, 0, y[, 1L] * 0.8, y[, 2L] * 0.8, col = col[2L], length
## = arrow.len): zero-length arrow is of indeterminate angle and so skipped

## Warning in arrows(0, 0, y[, 1L] * 0.8, y[, 2L] * 0.8, col = col[2L], length
## = arrow.len): zero-length arrow is of indeterminate angle and so skipped

## Warning in arrows(0, 0, y[, 1L] * 0.8, y[, 2L] * 0.8, col = col[2L], length
## = arrow.len): zero-length arrow is of indeterminate angle and so skipped

human.std <- scale(human)
pca_human2 <- prcomp(human.std)
biplot(pca_human2, choices= 1:2, cex = c(0.8, 1), col= c("grey40", "deeppink2"), main="Graph 3. PCA with human data, scaled")

s <- summary(pca_human2)
pca_pr <- round(100*s$importance[2, ], digits = 1)
pc_lab <- paste0(names(pca_pr), " (", pca_pr, "%)")
biplot(pca_human2, cex = c(0.8, 1), col = c("grey40", "deeppink2"), xlab = pc_lab[1], ylab = pc_lab[2], main="Graph 4. PCA with human data, scaled, %")

library(FactoMineR)
es.pca = PCA(human, scale.unit= TRUE, ncp=5)

tea dataset

library(ggplot2)
library(dplyr)
library(tidyr)
library(FactoMineR)
data(tea)
my_tea <- data.frame(tea)
dim(my_tea)
## [1] 300  36
str(my_tea)
## 'data.frame':    300 obs. of  36 variables:
##  $ breakfast       : Factor w/ 2 levels "breakfast","Not.breakfast": 1 1 2 2 1 2 1 2 1 1 ...
##  $ tea.time        : Factor w/ 2 levels "Not.tea time",..: 1 1 2 1 1 1 2 2 2 1 ...
##  $ evening         : Factor w/ 2 levels "evening","Not.evening": 2 2 1 2 1 2 2 1 2 1 ...
##  $ lunch           : Factor w/ 2 levels "lunch","Not.lunch": 2 2 2 2 2 2 2 2 2 2 ...
##  $ dinner          : Factor w/ 2 levels "dinner","Not.dinner": 2 2 1 1 2 1 2 2 2 2 ...
##  $ always          : Factor w/ 2 levels "always","Not.always": 2 2 2 2 1 2 2 2 2 2 ...
##  $ home            : Factor w/ 2 levels "home","Not.home": 1 1 1 1 1 1 1 1 1 1 ...
##  $ work            : Factor w/ 2 levels "Not.work","work": 1 1 2 1 1 1 1 1 1 1 ...
##  $ tearoom         : Factor w/ 2 levels "Not.tearoom",..: 1 1 1 1 1 1 1 1 1 2 ...
##  $ friends         : Factor w/ 2 levels "friends","Not.friends": 2 2 1 2 2 2 1 2 2 2 ...
##  $ resto           : Factor w/ 2 levels "Not.resto","resto": 1 1 2 1 1 1 1 1 1 1 ...
##  $ pub             : Factor w/ 2 levels "Not.pub","pub": 1 1 1 1 1 1 1 1 1 1 ...
##  $ Tea             : Factor w/ 3 levels "black","Earl Grey",..: 1 1 2 2 2 2 2 1 2 1 ...
##  $ How             : Factor w/ 4 levels "alone","lemon",..: 1 3 1 1 1 1 1 3 3 1 ...
##  $ sugar           : Factor w/ 2 levels "No.sugar","sugar": 2 1 1 2 1 1 1 1 1 1 ...
##  $ how             : Factor w/ 3 levels "tea bag","tea bag+unpackaged",..: 1 1 1 1 1 1 1 1 2 2 ...
##  $ where           : Factor w/ 3 levels "chain store",..: 1 1 1 1 1 1 1 1 2 2 ...
##  $ price           : Factor w/ 6 levels "p_branded","p_cheap",..: 4 6 6 6 6 3 6 6 5 5 ...
##  $ age             : int  39 45 47 23 48 21 37 36 40 37 ...
##  $ sex             : Factor w/ 2 levels "F","M": 2 1 1 2 2 2 2 1 2 2 ...
##  $ SPC             : Factor w/ 7 levels "employee","middle",..: 2 2 4 6 1 6 5 2 5 5 ...
##  $ Sport           : Factor w/ 2 levels "Not.sportsman",..: 2 2 2 1 2 2 2 2 2 1 ...
##  $ age_Q           : Factor w/ 5 levels "15-24","25-34",..: 3 4 4 1 4 1 3 3 3 3 ...
##  $ frequency       : Factor w/ 4 levels "1/day","1 to 2/week",..: 1 1 3 1 3 1 4 2 3 3 ...
##  $ escape.exoticism: Factor w/ 2 levels "escape-exoticism",..: 2 1 2 1 1 2 2 2 2 2 ...
##  $ spirituality    : Factor w/ 2 levels "Not.spirituality",..: 1 1 1 2 2 1 1 1 1 1 ...
##  $ healthy         : Factor w/ 2 levels "healthy","Not.healthy": 1 1 1 1 2 1 1 1 2 1 ...
##  $ diuretic        : Factor w/ 2 levels "diuretic","Not.diuretic": 2 1 1 2 1 2 2 2 2 1 ...
##  $ friendliness    : Factor w/ 2 levels "friendliness",..: 2 2 1 2 1 2 2 1 2 1 ...
##  $ iron.absorption : Factor w/ 2 levels "iron absorption",..: 2 2 2 2 2 2 2 2 2 2 ...
##  $ feminine        : Factor w/ 2 levels "feminine","Not.feminine": 2 2 2 2 2 2 2 1 2 2 ...
##  $ sophisticated   : Factor w/ 2 levels "Not.sophisticated",..: 1 1 1 2 1 1 1 2 2 1 ...
##  $ slimming        : Factor w/ 2 levels "No.slimming",..: 1 1 1 1 1 1 1 1 1 1 ...
##  $ exciting        : Factor w/ 2 levels "exciting","No.exciting": 2 1 2 2 2 2 2 2 2 2 ...
##  $ relaxing        : Factor w/ 2 levels "No.relaxing",..: 1 1 2 2 2 2 2 2 2 2 ...
##  $ effect.on.health: Factor w/ 2 levels "effect on health",..: 2 2 2 2 2 2 2 2 2 2 ...
summary(my_tea)
##          breakfast           tea.time          evening          lunch    
##  breakfast    :144   Not.tea time:131   evening    :103   lunch    : 44  
##  Not.breakfast:156   tea time    :169   Not.evening:197   Not.lunch:256  
##                                                                          
##                                                                          
##                                                                          
##                                                                          
##                                                                          
##         dinner           always          home           work    
##  dinner    : 21   always    :103   home    :291   Not.work:213  
##  Not.dinner:279   Not.always:197   Not.home:  9   work    : 87  
##                                                                 
##                                                                 
##                                                                 
##                                                                 
##                                                                 
##         tearoom           friends          resto          pub     
##  Not.tearoom:242   friends    :196   Not.resto:221   Not.pub:237  
##  tearoom    : 58   Not.friends:104   resto    : 79   pub    : 63  
##                                                                   
##                                                                   
##                                                                   
##                                                                   
##                                                                   
##         Tea         How           sugar                     how     
##  black    : 74   alone:195   No.sugar:155   tea bag           :170  
##  Earl Grey:193   lemon: 33   sugar   :145   tea bag+unpackaged: 94  
##  green    : 33   milk : 63                  unpackaged        : 36  
##                  other:  9                                          
##                                                                     
##                                                                     
##                                                                     
##                   where                 price          age        sex    
##  chain store         :192   p_branded      : 95   Min.   :15.00   F:178  
##  chain store+tea shop: 78   p_cheap        :  7   1st Qu.:23.00   M:122  
##  tea shop            : 30   p_private label: 21   Median :32.00          
##                             p_unknown      : 12   Mean   :37.05          
##                             p_upscale      : 53   3rd Qu.:48.00          
##                             p_variable     :112   Max.   :90.00          
##                                                                          
##            SPC               Sport       age_Q          frequency  
##  employee    :59   Not.sportsman:121   15-24:92   1/day      : 95  
##  middle      :40   sportsman    :179   25-34:69   1 to 2/week: 44  
##  non-worker  :64                       35-44:40   +2/day     :127  
##  other worker:20                       45-59:61   3 to 6/week: 34  
##  senior      :35                       +60  :38                    
##  student     :70                                                   
##  workman     :12                                                   
##              escape.exoticism           spirituality        healthy   
##  escape-exoticism    :142     Not.spirituality:206   healthy    :210  
##  Not.escape-exoticism:158     spirituality    : 94   Not.healthy: 90  
##                                                                       
##                                                                       
##                                                                       
##                                                                       
##                                                                       
##          diuretic             friendliness            iron.absorption
##  diuretic    :174   friendliness    :242   iron absorption    : 31   
##  Not.diuretic:126   Not.friendliness: 58   Not.iron absorption:269   
##                                                                      
##                                                                      
##                                                                      
##                                                                      
##                                                                      
##          feminine             sophisticated        slimming  
##  feminine    :129   Not.sophisticated: 85   No.slimming:255  
##  Not.feminine:171   sophisticated    :215   slimming   : 45  
##                                                              
##                                                              
##                                                              
##                                                              
##                                                              
##         exciting          relaxing              effect.on.health
##  exciting   :116   No.relaxing:113   effect on health   : 66    
##  No.exciting:184   relaxing   :187   No.effect on health:234    
##                                                                 
##                                                                 
##                                                                 
##                                                                 
## 
gather(my_tea) %>% ggplot(aes(value))  + facet_wrap("key", scales = "free") + geom_bar() + theme(axis.text.x = element_text(angle = 45, hjust = 1, size = 8))
## Warning: attributes are not identical across measure variables;
## they will be dropped

keep_columns <- c("breakfast", "sex", "price", "healthy", "spirituality", "friends", "Tea", "tearoom")
my_tea1 <- dplyr::select(my_tea, one_of(keep_columns))
gather(my_tea1) %>% ggplot(aes(value)) + facet_wrap("key", scales = "free") + geom_bar() + theme(axis.text.x = element_text(angle = 45, hjust = 1, size = 8))
## Warning: attributes are not identical across measure variables;
## they will be dropped

3. Multiple Correspondence Analysis

mca <- MCA(my_tea1, graph = FALSE)
summary(mca)
## 
## Call:
## MCA(X = my_tea1, graph = FALSE) 
## 
## 
## Eigenvalues
##                        Dim.1   Dim.2   Dim.3   Dim.4   Dim.5   Dim.6
## Variance               0.193   0.165   0.150   0.147   0.139   0.128
## % of var.             11.885  10.164   9.218   9.031   8.544   7.859
## Cumulative % of var.  11.885  22.050  31.268  40.299  48.843  56.702
##                        Dim.7   Dim.8   Dim.9  Dim.10  Dim.11  Dim.12
## Variance               0.121   0.115   0.110   0.102   0.096   0.087
## % of var.              7.429   7.054   6.784   6.297   5.928   5.371
## Cumulative % of var.  64.131  71.185  77.970  84.267  90.195  95.565
##                       Dim.13
## Variance               0.072
## % of var.              4.435
## Cumulative % of var. 100.000
## 
## Individuals (the 10 first)
##                    Dim.1    ctr   cos2    Dim.2    ctr   cos2    Dim.3
## 1               |  0.767  1.015  0.144 |  0.347  0.242  0.029 |  0.713
## 2               |  0.261  0.117  0.057 |  0.285  0.164  0.069 | -0.327
## 3               | -0.400  0.276  0.233 | -0.236  0.113  0.081 |  0.005
## 4               |  0.057  0.006  0.003 | -0.199  0.080  0.034 |  0.209
## 5               | -0.005  0.000  0.000 |  0.067  0.009  0.003 |  0.484
## 6               |  0.641  0.709  0.171 | -0.080  0.013  0.003 |  0.234
## 7               | -0.151  0.039  0.028 |  0.019  0.001  0.000 |  0.020
## 8               |  0.262  0.118  0.059 |  0.161  0.052  0.022 | -0.149
## 9               |  0.533  0.491  0.180 |  0.465  0.437  0.137 |  0.375
## 10              |  0.503  0.436  0.118 |  1.130  2.577  0.594 | -0.477
##                    ctr   cos2  
## 1                1.132  0.125 |
## 2                0.238  0.090 |
## 3                0.000  0.000 |
## 4                0.097  0.037 |
## 5                0.521  0.164 |
## 6                0.122  0.023 |
## 7                0.001  0.001 |
## 8                0.050  0.019 |
## 9                0.313  0.089 |
## 10               0.505  0.106 |
## 
## Categories (the 10 first)
##                     Dim.1     ctr    cos2  v.test     Dim.2     ctr
## breakfast       |  -0.002   0.000   0.000  -0.030 |   0.211   1.613
## Not.breakfast   |   0.002   0.000   0.000   0.030 |  -0.194   1.489
## F               |  -0.357   4.897   0.186  -7.458 |  -0.173   1.347
## M               |   0.521   7.144   0.186   7.458 |   0.253   1.965
## p_branded       |   0.289   1.708   0.039   3.398 |  -0.960  22.087
## p_cheap         |   0.678   0.694   0.011   1.812 |   0.329   0.191
## p_private label |   0.723   2.369   0.039   3.431 |   0.274   0.398
## p_unknown       |   0.192   0.095   0.002   0.677 |  -0.040   0.005
## p_upscale       |   0.563   3.621   0.068   4.507 |   1.183  18.709
## p_variable      |  -0.710  12.168   0.300  -9.471 |   0.187   0.986
##                    cos2  v.test     Dim.3     ctr    cos2  v.test  
## breakfast         0.041   3.500 |  -0.286   3.276   0.075  -4.751 |
## Not.breakfast     0.041  -3.500 |   0.264   3.024   0.075   4.751 |
## F                 0.044  -3.618 |  -0.242   2.905   0.086  -5.059 |
## M                 0.044   3.618 |   0.353   4.238   0.086   5.059 |
## p_branded         0.427 -11.300 |  -0.341   3.069   0.054  -4.012 |
## p_cheap           0.003   0.880 |   1.264   3.110   0.038   3.378 |
## p_private label   0.006   1.301 |   0.150   0.131   0.002   0.709 |
## p_unknown         0.000  -0.142 |   2.649  23.430   0.292   9.351 |
## p_upscale         0.300   9.475 |  -0.267   1.049   0.015  -2.137 |
## p_variable        0.021   2.493 |   0.024   0.019   0.000   0.326 |
## 
## Categorical variables (eta2)
##                   Dim.1 Dim.2 Dim.3  
## breakfast       | 0.000 0.041 0.075 |
## sex             | 0.186 0.044 0.086 |
## price           | 0.319 0.560 0.369 |
## healthy         | 0.010 0.044 0.413 |
## spirituality    | 0.082 0.019 0.000 |
## friends         | 0.412 0.000 0.000 |
## Tea             | 0.272 0.340 0.162 |
## tearoom         | 0.264 0.273 0.092 |
plot(mca, invisible=c("ind"),  habillage="quali")

res.mca=MCA(my_tea1)


Analysis of longitudinal data (Exercise 6)

Read data

BPRS<-read.table("https://raw.githubusercontent.com/KimmoVehkalahti/MABS/master/Examples/data/BPRS.txt", sep=" ", header=TRUE)
RATS<- read.table("https://raw.githubusercontent.com/KimmoVehkalahti/MABS/master/Examples/data/rats.txt", sep="\t", header=TRUE)
library(ggplot2);library(dplyr);library(tidyr)

Task 1

Convert catergorial variables in RATS dataset to factors and to long form

RATS$Group<-factor(RATS$Group)
RATS$ID<-factor(RATS$ID)
glimpse(RATS)
## Observations: 16
## Variables: 13
## $ ID    <fct> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16
## $ Group <fct> 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3
## $ WD1   <int> 240, 225, 245, 260, 255, 260, 275, 245, 410, 405, 445, 5...
## $ WD8   <int> 250, 230, 250, 255, 260, 265, 275, 255, 415, 420, 445, 5...
## $ WD15  <int> 255, 230, 250, 255, 255, 270, 260, 260, 425, 430, 450, 5...
## $ WD22  <int> 260, 232, 255, 265, 270, 275, 270, 268, 428, 440, 452, 5...
## $ WD29  <int> 262, 240, 262, 265, 270, 275, 273, 270, 438, 448, 455, 5...
## $ WD36  <int> 258, 240, 265, 268, 273, 277, 274, 265, 443, 460, 455, 5...
## $ WD43  <int> 266, 243, 267, 270, 274, 278, 276, 265, 442, 458, 451, 5...
## $ WD44  <int> 266, 244, 267, 272, 273, 278, 271, 267, 446, 464, 450, 5...
## $ WD50  <int> 265, 238, 264, 274, 276, 284, 282, 273, 456, 475, 462, 6...
## $ WD57  <int> 272, 247, 268, 273, 278, 279, 281, 274, 468, 484, 466, 6...
## $ WD64  <int> 278, 245, 269, 275, 280, 281, 284, 278, 478, 496, 472, 6...
head(RATS)
##   ID Group WD1 WD8 WD15 WD22 WD29 WD36 WD43 WD44 WD50 WD57 WD64
## 1  1     1 240 250  255  260  262  258  266  266  265  272  278
## 2  2     1 225 230  230  232  240  240  243  244  238  247  245
## 3  3     1 245 250  250  255  262  265  267  267  264  268  269
## 4  4     1 260 255  255  265  265  268  270  272  274  273  275
## 5  5     1 255 260  255  270  270  273  274  273  276  278  280
## 6  6     1 260 265  270  275  275  277  278  278  284  279  281
RATSL<-gather(RATS, key=WDs, value=weight, WD1:WD64)%>%mutate(time= as.integer(substr(WDs,3,4)))
glimpse(RATSL)
## Observations: 176
## Variables: 5
## $ ID     <fct> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, ...
## $ Group  <fct> 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 1, 1, 1...
## $ WDs    <chr> "WD1", "WD1", "WD1", "WD1", "WD1", "WD1", "WD1", "WD1",...
## $ weight <int> 240, 225, 245, 260, 255, 260, 275, 245, 410, 405, 445, ...
## $ time   <int> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 8, 8...
head(RATSL)
##   ID Group WDs weight time
## 1  1     1 WD1    240    1
## 2  2     1 WD1    225    1
## 3  3     1 WD1    245    1
## 4  4     1 WD1    260    1
## 5  5     1 WD1    255    1
## 6  6     1 WD1    260    1
tail(RATSL)
##     ID Group  WDs weight time
## 171 11     2 WD64    472   64
## 172 12     2 WD64    628   64
## 173 13     3 WD64    525   64
## 174 14     3 WD64    559   64
## 175 15     3 WD64    548   64
## 176 16     3 WD64    569   64
ggplot(RATSL, aes(x = time, y = weight, linetype = ID)) +
  geom_line() +
  scale_linetype_manual(values = rep(1:10, times=4)) +
  facet_grid(. ~ Group, labeller = label_both) +
  theme(legend.position = "none") + 
  scale_y_continuous(limits = c(min(RATSL$weight), max(RATSL$weight)))

Standardise the weight

RATSL<- RATSL%>%group_by(time)%>%mutate(stdweight = (weight - mean(weight))/sd(weight))%>%ungroup()
glimpse(RATSL)
## Observations: 176
## Variables: 6
## $ ID        <fct> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 1...
## $ Group     <fct> 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 1, 1...
## $ WDs       <chr> "WD1", "WD1", "WD1", "WD1", "WD1", "WD1", "WD1", "WD...
## $ weight    <int> 240, 225, 245, 260, 255, 260, 275, 245, 410, 405, 44...
## $ time      <int> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 8...
## $ stdweight <dbl> -1.0011429, -1.1203857, -0.9613953, -0.8421525, -0.8...
p1<-ggplot(RATSL, aes(x=time, y=stdweight, linetype=ID))
p2<-p1+geom_line()+scale_linetype_manual(values=rep(1:10, times=4))
p3<-p2+facet_grid(. ~Group, labeller = label_both)
p4<-p3+theme_bw()+theme(legend.position ="none")
p5<-p4+theme(panel.grid.minor.y=element_blank())
p6<-p5+scale_y_continuous(name="standardized weights")
p6

Numberof times, baseline (time 0)

n<-RATSL$time%>%unique()%>%length()

Make a summary data

RATSS<-RATSL%>%group_by(Group, time)%>% summarise(mean=mean(weight), se=sd(weight)/sqrt(n))%>%ungroup()
glimpse(RATSS)
## Observations: 33
## Variables: 4
## $ Group <fct> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2,...
## $ time  <int> 1, 8, 15, 22, 29, 36, 43, 44, 50, 57, 64, 1, 8, 15, 22, ...
## $ mean  <dbl> 250.625, 255.000, 254.375, 261.875, 264.625, 265.000, 26...
## $ se    <dbl> 4.589478, 3.947710, 3.460116, 4.100800, 3.333956, 3.5529...

Visualize the mean weight and SE

p1<-ggplot(RATSS, aes(x=time, y=mean, linetype=Group, shape=Group))
p2<-p1+geom_line()+scale_linetype_manual(values=c(1,2,3))
p3<-p2+geom_point(size=3)+scale_shape_manual(values=c(1,2,3))
p4<-p3+geom_errorbar(aes(ymin=mean-se, ymax=mean+se, linetype="1"), width=0.3)
p5<-p4+theme_bw()+theme(panel.grid.major=element_blank(), panel.grid.minor=element_blank())
p6<-p5+theme(legend.position= "top")
p7<-p6+scale_y_continuous(name="mean(weights) +/- se(weights)")
p7

Task 2

BPRS$treatment<-factor(BPRS$treatment)
BPRS$subject<-factor(BPRS$subject)
str(BPRS)
## 'data.frame':    40 obs. of  11 variables:
##  $ treatment: Factor w/ 2 levels "1","2": 1 1 1 1 1 1 1 1 1 1 ...
##  $ subject  : Factor w/ 20 levels "1","2","3","4",..: 1 2 3 4 5 6 7 8 9 10 ...
##  $ week0    : int  42 58 54 55 72 48 71 30 41 57 ...
##  $ week1    : int  36 68 55 77 75 43 61 36 43 51 ...
##  $ week2    : int  36 61 41 49 72 41 47 38 39 51 ...
##  $ week3    : int  43 55 38 54 65 38 30 38 35 55 ...
##  $ week4    : int  41 43 43 56 50 36 27 31 28 53 ...
##  $ week5    : int  40 34 28 50 39 29 40 26 22 43 ...
##  $ week6    : int  38 28 29 47 32 33 30 26 20 43 ...
##  $ week7    : int  47 28 25 42 38 27 31 25 23 39 ...
##  $ week8    : int  51 28 24 46 32 25 31 24 21 32 ...
BPRSL<-gather(BPRS, key=weeks, value=bprs, -treatment, -subject)%>%mutate(week=as.integer(substr(weeks, 5, 5)))
glimpse(BPRSL)
## Observations: 360
## Variables: 5
## $ treatment <fct> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...
## $ subject   <fct> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 1...
## $ weeks     <chr> "week0", "week0", "week0", "week0", "week0", "week0"...
## $ bprs      <int> 42, 58, 54, 55, 72, 48, 71, 30, 41, 57, 30, 55, 36, ...
## $ week      <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...
head(BPRSL)
##   treatment subject weeks bprs week
## 1         1       1 week0   42    0
## 2         1       2 week0   58    0
## 3         1       3 week0   54    0
## 4         1       4 week0   55    0
## 5         1       5 week0   72    0
## 6         1       6 week0   48    0
tail(BPRSL)
##     treatment subject weeks bprs week
## 355         2      15 week8   37    8
## 356         2      16 week8   22    8
## 357         2      17 week8   43    8
## 358         2      18 week8   30    8
## 359         2      19 week8   21    8
## 360         2      20 week8   23    8
BPRS_reg<-lm(bprs ~ week + treatment, data=BPRSL)
summary(BPRS_reg)
## 
## Call:
## lm(formula = bprs ~ week + treatment, data = BPRSL)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -22.454  -8.965  -3.196   7.002  50.244 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  46.4539     1.3670  33.982   <2e-16 ***
## week         -2.2704     0.2524  -8.995   <2e-16 ***
## treatment2    0.5722     1.3034   0.439    0.661    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 12.37 on 357 degrees of freedom
## Multiple R-squared:  0.1851, Adjusted R-squared:  0.1806 
## F-statistic: 40.55 on 2 and 357 DF,  p-value: < 2.2e-16
library(lme4)
## Loading required package: Matrix
## 
## Attaching package: 'Matrix'
## The following objects are masked from 'package:tidyr':
## 
##     expand, pack, unpack
BPRS_ref<-lmer(bprs ~ week + treatment + (1|subject), data=BPRSL, REML=F)
summary(BPRS_ref)
## Linear mixed model fit by maximum likelihood  ['lmerMod']
## Formula: bprs ~ week + treatment + (1 | subject)
##    Data: BPRSL
## 
##      AIC      BIC   logLik deviance df.resid 
##   2748.7   2768.1  -1369.4   2738.7      355 
## 
## Scaled residuals: 
##     Min      1Q  Median      3Q     Max 
## -3.0481 -0.6749 -0.1361  0.4813  3.4855 
## 
## Random effects:
##  Groups   Name        Variance Std.Dev.
##  subject  (Intercept)  47.41    6.885  
##  Residual             104.21   10.208  
## Number of obs: 360, groups:  subject, 20
## 
## Fixed effects:
##             Estimate Std. Error t value
## (Intercept)  46.4539     1.9090  24.334
## week         -2.2704     0.2084 -10.896
## treatment2    0.5722     1.0761   0.532
## 
## Correlation of Fixed Effects:
##            (Intr) week  
## week       -0.437       
## treatment2 -0.282  0.000
BPRS_ref1<-lmer(bprs~week+treatment+(week|subject), data=BPRSL, REML=F)
summary(BPRS_ref1)
## Linear mixed model fit by maximum likelihood  ['lmerMod']
## Formula: bprs ~ week + treatment + (week | subject)
##    Data: BPRSL
## 
##      AIC      BIC   logLik deviance df.resid 
##   2745.4   2772.6  -1365.7   2731.4      353 
## 
## Scaled residuals: 
##     Min      1Q  Median      3Q     Max 
## -2.8919 -0.6194 -0.0691  0.5531  3.7977 
## 
## Random effects:
##  Groups   Name        Variance Std.Dev. Corr 
##  subject  (Intercept) 64.8222  8.0512        
##           week         0.9609  0.9803   -0.51
##  Residual             97.4304  9.8707        
## Number of obs: 360, groups:  subject, 20
## 
## Fixed effects:
##             Estimate Std. Error t value
## (Intercept)  46.4539     2.1052  22.066
## week         -2.2704     0.2977  -7.626
## treatment2    0.5722     1.0405   0.550
## 
## Correlation of Fixed Effects:
##            (Intr) week  
## week       -0.582       
## treatment2 -0.247  0.000
anova(BPRS_ref1, BPRS_ref)
## Data: BPRSL
## Models:
## BPRS_ref: bprs ~ week + treatment + (1 | subject)
## BPRS_ref1: bprs ~ week + treatment + (week | subject)
##           Df    AIC    BIC  logLik deviance  Chisq Chi Df Pr(>Chisq)  
## BPRS_ref   5 2748.7 2768.1 -1369.4   2738.7                           
## BPRS_ref1  7 2745.4 2772.6 -1365.7   2731.4 7.2721      2    0.02636 *
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
BPRS_ref2<-lmer(bprs~week+treatment+(treatment|subject), data=BPRSL, REML=F)
summary(BPRS_ref2)
## Linear mixed model fit by maximum likelihood  ['lmerMod']
## Formula: bprs ~ week + treatment + (treatment | subject)
##    Data: BPRSL
## 
##      AIC      BIC   logLik deviance df.resid 
##   2584.7   2611.9  -1285.4   2570.7      353 
## 
## Scaled residuals: 
##     Min      1Q  Median      3Q     Max 
## -2.3048 -0.6022 -0.0652  0.4414  3.1690 
## 
## Random effects:
##  Groups   Name        Variance Std.Dev. Corr 
##  subject  (Intercept)  64.43    8.027        
##           treatment2  188.83   13.742   -0.56
##  Residual              54.23    7.364        
## Number of obs: 360, groups:  subject, 20
## 
## Fixed effects:
##             Estimate Std. Error t value
## (Intercept)  46.4539     1.9708  23.571
## week         -2.2704     0.1503 -15.104
## treatment2    0.5722     3.1692   0.181
## 
## Correlation of Fixed Effects:
##            (Intr) week  
## week       -0.305       
## treatment2 -0.540  0.000
anova(BPRS_ref1, BPRS_ref2)
## Data: BPRSL
## Models:
## BPRS_ref1: bprs ~ week + treatment + (week | subject)
## BPRS_ref2: bprs ~ week + treatment + (treatment | subject)
##           Df    AIC    BIC  logLik deviance Chisq Chi Df Pr(>Chisq)    
## BPRS_ref1  7 2745.4 2772.6 -1365.7   2731.4                            
## BPRS_ref2  7 2584.7 2611.9 -1285.4   2570.7 160.7      0  < 2.2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Fitted<-fitted(BPRS_ref2)
BPRSL%>% mutate(Fitted)
##     treatment subject weeks bprs week   Fitted
## 1           1       1 week0   42    0 50.29960
## 2           1       2 week0   58    0 53.19735
## 3           1       3 week0   54    0 46.50943
## 4           1       4 week0   55    0 60.62454
## 5           1       5 week0   72    0 60.56119
## 6           1       6 week0   48    0 44.77537
## 7           1       7 week0   71    0 49.67670
## 8           1       8 week0   30    0 40.15001
## 9           1       9 week0   41    0 39.89941
## 10          1      10 week0   57    0 55.36484
## 11          1      11 week0   30    0 44.95909
## 12          1      12 week0   55    0 53.41261
## 13          1      13 week0   36    0 37.39683
## 14          1      14 week0   38    0 39.90752
## 15          1      15 week0   66    0 53.93870
## 16          1      16 week0   41    0 43.76270
## 17          1      17 week0   45    0 45.30333
## 18          1      18 week0   39    0 37.56531
## 19          1      19 week0   24    0 35.43005
## 20          1      20 week0   38    0 36.34320
## 21          2       1 week0   52    0 57.49443
## 22          2       2 week0   30    0 32.65015
## 23          2       3 week0   65    0 41.66325
## 24          2       4 week0   37    0 37.97522
## 25          2       5 week0   59    0 57.51543
## 26          2       6 week0   30    0 38.15519
## 27          2       7 week0   69    0 50.69652
## 28          2       8 week0   62    0 62.78354
## 29          2       9 week0   38    0 38.57000
## 30          2      10 week0   65    0 49.96478
## 31          2      11 week0   78    0 80.10357
## 32          2      12 week0   38    0 38.81005
## 33          2      13 week0   63    0 57.68042
## 34          2      14 week0   40    0 42.71172
## 35          2      15 week0   40    0 48.05030
## 36          2      16 week0   54    0 39.85227
## 37          2      17 week0   33    0 48.35123
## 38          2      18 week0   28    0 39.94579
## 39          2      19 week0   52    0 39.09184
## 40          2      20 week0   47    0 38.45652
## 41          1       1 week1   36    1 48.02919
## 42          1       2 week1   68    1 50.92693
## 43          1       3 week1   55    1 44.23901
## 44          1       4 week1   77    1 58.35412
## 45          1       5 week1   75    1 58.29078
## 46          1       6 week1   43    1 42.50495
## 47          1       7 week1   61    1 47.40628
## 48          1       8 week1   36    1 37.87959
## 49          1       9 week1   43    1 37.62899
## 50          1      10 week1   51    1 53.09442
## 51          1      11 week1   34    1 42.68868
## 52          1      12 week1   52    1 51.14219
## 53          1      13 week1   32    1 35.12642
## 54          1      14 week1   35    1 37.63710
## 55          1      15 week1   68    1 51.66828
## 56          1      16 week1   35    1 41.49228
## 57          1      17 week1   38    1 43.03291
## 58          1      18 week1   35    1 35.29490
## 59          1      19 week1   28    1 33.15963
## 60          1      20 week1   34    1 34.07279
## 61          2       1 week1   73    1 55.22402
## 62          2       2 week1   23    1 30.37974
## 63          2       3 week1   31    1 39.39284
## 64          2       4 week1   31    1 35.70481
## 65          2       5 week1   67    1 55.24501
## 66          2       6 week1   33    1 35.88477
## 67          2       7 week1   52    1 48.42610
## 68          2       8 week1   54    1 60.51312
## 69          2       9 week1   40    1 36.29958
## 70          2      10 week1   44    1 47.69436
## 71          2      11 week1   95    1 77.83315
## 72          2      12 week1   41    1 36.53963
## 73          2      13 week1   65    1 55.41001
## 74          2      14 week1   37    1 40.44130
## 75          2      15 week1   36    1 45.77989
## 76          2      16 week1   45    1 37.58186
## 77          2      17 week1   41    1 46.08081
## 78          2      18 week1   30    1 37.67537
## 79          2      19 week1   43    1 36.82142
## 80          2      20 week1   36    1 36.18611
## 81          1       1 week2   36    2 45.75877
## 82          1       2 week2   61    2 48.65652
## 83          1       3 week2   41    2 41.96860
## 84          1       4 week2   49    2 56.08371
## 85          1       5 week2   72    2 56.02036
## 86          1       6 week2   41    2 40.23454
## 87          1       7 week2   47    2 45.13587
## 88          1       8 week2   38    2 35.60918
## 89          1       9 week2   39    2 35.35857
## 90          1      10 week2   51    2 50.82400
## 91          1      11 week2   34    2 40.41826
## 92          1      12 week2   49    2 48.87178
## 93          1      13 week2   36    2 32.85600
## 94          1      14 week2   36    2 35.36668
## 95          1      15 week2   65    2 49.39786
## 96          1      16 week2   45    2 39.22186
## 97          1      17 week2   46    2 40.76250
## 98          1      18 week2   27    2 33.02448
## 99          1      19 week2   31    2 30.88922
## 100         1      20 week2   27    2 31.80237
## 101         2       1 week2   42    2 52.95360
## 102         2       2 week2   32    2 28.10932
## 103         2       3 week2   33    2 37.12242
## 104         2       4 week2   27    2 33.43439
## 105         2       5 week2   58    2 52.97460
## 106         2       6 week2   37    2 33.61435
## 107         2       7 week2   41    2 46.15569
## 108         2       8 week2   49    2 58.24270
## 109         2       9 week2   38    2 34.02917
## 110         2      10 week2   31    2 45.42394
## 111         2      11 week2   75    2 75.56273
## 112         2      12 week2   36    2 34.26921
## 113         2      13 week2   60    2 53.13959
## 114         2      14 week2   31    2 38.17088
## 115         2      15 week2   55    2 43.50947
## 116         2      16 week2   35    2 35.31144
## 117         2      17 week2   30    2 43.81039
## 118         2      18 week2   29    2 35.40496
## 119         2      19 week2   26    2 34.55101
## 120         2      20 week2   32    2 33.91569
## 121         1       1 week3   43    3 43.48835
## 122         1       2 week3   55    3 46.38610
## 123         1       3 week3   38    3 39.69818
## 124         1       4 week3   54    3 53.81329
## 125         1       5 week3   65    3 53.74994
## 126         1       6 week3   38    3 37.96412
## 127         1       7 week3   30    3 42.86545
## 128         1       8 week3   38    3 33.33876
## 129         1       9 week3   35    3 33.08816
## 130         1      10 week3   55    3 48.55359
## 131         1      11 week3   41    3 38.14784
## 132         1      12 week3   54    3 46.60136
## 133         1      13 week3   31    3 30.58558
## 134         1      14 week3   34    3 33.09627
## 135         1      15 week3   49    3 47.12745
## 136         1      16 week3   42    3 36.95145
## 137         1      17 week3   38    3 38.49208
## 138         1      18 week3   25    3 30.75406
## 139         1      19 week3   28    3 28.61880
## 140         1      20 week3   25    3 29.53195
## 141         2       1 week3   41    3 50.68318
## 142         2       2 week3   24    3 25.83890
## 143         2       3 week3   28    3 34.85200
## 144         2       4 week3   31    3 31.16397
## 145         2       5 week3   61    3 50.70418
## 146         2       6 week3   33    3 31.34394
## 147         2       7 week3   33    3 43.88527
## 148         2       8 week3   39    3 55.97229
## 149         2       9 week3   27    3 31.75875
## 150         2      10 week3   34    3 43.15353
## 151         2      11 week3   76    3 73.29232
## 152         2      12 week3   27    3 31.99880
## 153         2      13 week3   53    3 50.86917
## 154         2      14 week3   38    3 35.90047
## 155         2      15 week3   55    3 41.23905
## 156         2      16 week3   27    3 33.04102
## 157         2      17 week3   32    3 41.53998
## 158         2      18 week3   33    3 33.13454
## 159         2      19 week3   27    3 32.28059
## 160         2      20 week3   29    3 31.64527
## 161         1       1 week4   41    4 41.21794
## 162         1       2 week4   43    4 44.11568
## 163         1       3 week4   43    4 37.42776
## 164         1       4 week4   56    4 51.54287
## 165         1       5 week4   50    4 51.47953
## 166         1       6 week4   36    4 35.69370
## 167         1       7 week4   27    4 40.59503
## 168         1       8 week4   31    4 31.06834
## 169         1       9 week4   28    4 30.81774
## 170         1      10 week4   53    4 46.28317
## 171         1      11 week4   36    4 35.87743
## 172         1      12 week4   48    4 44.33094
## 173         1      13 week4   25    4 28.31517
## 174         1      14 week4   25    4 30.82585
## 175         1      15 week4   36    4 44.85703
## 176         1      16 week4   31    4 34.68103
## 177         1      17 week4   40    4 36.22166
## 178         1      18 week4   29    4 28.48365
## 179         1      19 week4   29    4 26.34838
## 180         1      20 week4   25    4 27.26154
## 181         2       1 week4   39    4 48.41277
## 182         2       2 week4   20    4 23.56849
## 183         2       3 week4   22    4 32.58159
## 184         2       4 week4   31    4 28.89356
## 185         2       5 week4   49    4 48.43376
## 186         2       6 week4   28    4 29.07352
## 187         2       7 week4   34    4 41.61485
## 188         2       8 week4   55    4 53.70187
## 189         2       9 week4   31    4 29.48833
## 190         2      10 week4   39    4 40.88311
## 191         2      11 week4   66    4 71.02190
## 192         2      12 week4   29    4 29.72838
## 193         2      13 week4   52    4 48.59876
## 194         2      14 week4   35    4 33.63005
## 195         2      15 week4   42    4 38.96864
## 196         2      16 week4   25    4 30.77061
## 197         2      17 week4   46    4 39.26956
## 198         2      18 week4   30    4 30.86412
## 199         2      19 week4   24    4 30.01017
## 200         2      20 week4   25    4 29.37486
## 201         1       1 week5   40    5 38.94752
## 202         1       2 week5   34    5 41.84527
## 203         1       3 week5   28    5 35.15735
## 204         1       4 week5   50    5 49.27246
## 205         1       5 week5   39    5 49.20911
## 206         1       6 week5   29    5 33.42329
## 207         1       7 week5   40    5 38.32462
## 208         1       8 week5   26    5 28.79793
## 209         1       9 week5   22    5 28.54732
## 210         1      10 week5   43    5 44.01275
## 211         1      11 week5   36    5 33.60701
## 212         1      12 week5   43    5 42.06053
## 213         1      13 week5   25    5 26.04475
## 214         1      14 week5   27    5 28.55543
## 215         1      15 week5   32    5 42.58661
## 216         1      16 week5   31    5 32.41061
## 217         1      17 week5   33    5 33.95125
## 218         1      18 week5   28    5 26.21323
## 219         1      19 week5   21    5 24.07797
## 220         1      20 week5   27    5 24.99112
## 221         2       1 week5   38    5 46.14235
## 222         2       2 week5   20    5 21.29807
## 223         2       3 week5   25    5 30.31117
## 224         2       4 week5   26    5 26.62314
## 225         2       5 week5   38    5 46.16335
## 226         2       6 week5   26    5 26.80310
## 227         2       7 week5   37    5 39.34444
## 228         2       8 week5   51    5 51.43145
## 229         2       9 week5   24    5 27.21792
## 230         2      10 week5   34    5 38.61269
## 231         2      11 week5   64    5 68.75148
## 232         2      12 week5   27    5 27.45796
## 233         2      13 week5   32    5 46.32834
## 234         2      14 week5   30    5 31.35963
## 235         2      15 week5   30    5 36.69822
## 236         2      16 week5   22    5 28.50019
## 237         2      17 week5   43    5 36.99914
## 238         2      18 week5   26    5 28.59371
## 239         2      19 week5   32    5 27.73976
## 240         2      20 week5   23    5 27.10444
## 241         1       1 week6   38    6 36.67710
## 242         1       2 week6   28    6 39.57485
## 243         1       3 week6   29    6 32.88693
## 244         1       4 week6   47    6 47.00204
## 245         1       5 week6   32    6 46.93869
## 246         1       6 week6   33    6 31.15287
## 247         1       7 week6   30    6 36.05420
## 248         1       8 week6   26    6 26.52751
## 249         1       9 week6   20    6 26.27691
## 250         1      10 week6   43    6 41.74234
## 251         1      11 week6   38    6 31.33659
## 252         1      12 week6   37    6 39.79011
## 253         1      13 week6   21    6 23.77433
## 254         1      14 week6   25    6 26.28502
## 255         1      15 week6   27    6 40.31620
## 256         1      16 week6   29    6 30.14020
## 257         1      17 week6   27    6 31.68083
## 258         1      18 week6   21    6 23.94281
## 259         1      19 week6   22    6 21.80755
## 260         1      20 week6   21    6 22.72070
## 261         2       1 week6   43    6 43.87193
## 262         2       2 week6   19    6 19.02765
## 263         2       3 week6   24    6 28.04075
## 264         2       4 week6   24    6 24.35272
## 265         2       5 week6   37    6 43.89293
## 266         2       6 week6   27    6 24.53269
## 267         2       7 week6   37    6 37.07402
## 268         2       8 week6   55    6 49.16104
## 269         2       9 week6   22    6 24.94750
## 270         2      10 week6   41    6 36.34228
## 271         2      11 week6   64    6 66.48107
## 272         2      12 week6   21    6 25.18755
## 273         2      13 week6   37    6 44.05792
## 274         2      14 week6   33    6 29.08922
## 275         2      15 week6   26    6 34.42780
## 276         2      16 week6   22    6 26.22977
## 277         2      17 week6   43    6 34.72873
## 278         2      18 week6   36    6 26.32329
## 279         2      19 week6   21    6 25.46934
## 280         2      20 week6   23    6 24.83402
## 281         1       1 week7   47    7 34.40669
## 282         1       2 week7   28    7 37.30443
## 283         1       3 week7   25    7 30.61651
## 284         1       4 week7   42    7 44.73162
## 285         1       5 week7   38    7 44.66828
## 286         1       6 week7   27    7 28.88245
## 287         1       7 week7   31    7 33.78378
## 288         1       8 week7   25    7 24.25709
## 289         1       9 week7   23    7 24.00649
## 290         1      10 week7   39    7 39.47192
## 291         1      11 week7   36    7 29.06618
## 292         1      12 week7   36    7 37.51969
## 293         1      13 week7   19    7 21.50392
## 294         1      14 week7   26    7 24.01460
## 295         1      15 week7   30    7 38.04578
## 296         1      16 week7   26    7 27.86978
## 297         1      17 week7   31    7 29.41041
## 298         1      18 week7   25    7 21.67240
## 299         1      19 week7   23    7 19.53713
## 300         1      20 week7   19    7 20.45029
## 301         2       1 week7   62    7 41.60152
## 302         2       2 week7   18    7 16.75724
## 303         2       3 week7   31    7 25.77034
## 304         2       4 week7   26    7 22.08231
## 305         2       5 week7   36    7 41.62251
## 306         2       6 week7   23    7 22.26227
## 307         2       7 week7   38    7 34.80360
## 308         2       8 week7   59    7 46.89062
## 309         2       9 week7   21    7 22.67708
## 310         2      10 week7   42    7 34.07186
## 311         2      11 week7   60    7 64.21065
## 312         2      12 week7   22    7 22.91713
## 313         2      13 week7   52    7 41.78751
## 314         2      14 week7   30    7 26.81880
## 315         2      15 week7   30    7 32.15739
## 316         2      16 week7   22    7 23.95936
## 317         2      17 week7   43    7 32.45831
## 318         2      18 week7   33    7 24.05287
## 319         2      19 week7   21    7 23.19892
## 320         2      20 week7   23    7 22.56361
## 321         1       1 week8   51    8 32.13627
## 322         1       2 week8   28    8 35.03402
## 323         1       3 week8   24    8 28.34610
## 324         1       4 week8   46    8 42.46121
## 325         1       5 week8   32    8 42.39786
## 326         1       6 week8   25    8 26.61204
## 327         1       7 week8   31    8 31.51337
## 328         1       8 week8   24    8 21.98668
## 329         1       9 week8   21    8 21.73607
## 330         1      10 week8   32    8 37.20150
## 331         1      11 week8   36    8 26.79576
## 332         1      12 week8   31    8 35.24928
## 333         1      13 week8   22    8 19.23350
## 334         1      14 week8   26    8 21.74418
## 335         1      15 week8   37    8 35.77536
## 336         1      16 week8   30    8 25.59936
## 337         1      17 week8   27    8 27.14000
## 338         1      18 week8   20    8 19.40198
## 339         1      19 week8   22    8 17.26672
## 340         1      20 week8   21    8 18.17987
## 341         2       1 week8   50    8 39.33110
## 342         2       2 week8   20    8 14.48682
## 343         2       3 week8   32    8 23.49992
## 344         2       4 week8   23    8 19.81189
## 345         2       5 week8   35    8 39.35210
## 346         2       6 week8   21    8 19.99185
## 347         2       7 week8   35    8 32.53319
## 348         2       8 week8   66    8 44.62020
## 349         2       9 week8   21    8 20.40667
## 350         2      10 week8   39    8 31.80144
## 351         2      11 week8   75    8 61.94023
## 352         2      12 week8   23    8 20.64671
## 353         2      13 week8   28    8 39.51709
## 354         2      14 week8   27    8 24.54838
## 355         2      15 week8   37    8 29.88697
## 356         2      16 week8   22    8 21.68894
## 357         2      17 week8   43    8 30.18789
## 358         2      18 week8   30    8 21.78246
## 359         2      19 week8   21    8 20.92851
## 360         2      20 week8   23    8 20.29319